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Mirrors > Home > LLPE Home > Th. List > ilbd | Structured version |
Description: Construct a biconditional from its forward and reverse implications. |
Ref | Expression |
---|---|
ilbd.1 | ⊦ (𝜑 ⊸ (𝜓 ⊸ 𝜒)) |
ilbd.2 | ⊦ (𝜑 ⊸ (𝜒 ⊸ 𝜓)) |
Ref | Expression |
---|---|
ilbd | ⊦ (𝜑 ⊸ (𝜓 ⧟ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ilbd.1 | . . . . 5 ⊦ (𝜑 ⊸ (𝜓 ⊸ 𝜒)) | |
2 | 1 | dfli1i 65 | . . . 4 ⊦ (~ 𝜑 ⅋ (𝜓 ⊸ 𝜒)) |
3 | ilbd.2 | . . . . 5 ⊦ (𝜑 ⊸ (𝜒 ⊸ 𝜓)) | |
4 | 3 | dfli1i 65 | . . . 4 ⊦ (~ 𝜑 ⅋ (𝜒 ⊸ 𝜓)) |
5 | 2, 4 | ax-iac 32 | . . 3 ⊦ (~ 𝜑 ⅋ ((𝜓 ⊸ 𝜒) & (𝜒 ⊸ 𝜓))) |
6 | dflb 93 | . . 3 ⊦ ((𝜓 ⧟ 𝜒) ⧟ ((𝜓 ⊸ 𝜒) & (𝜒 ⊸ 𝜓))) | |
7 | 5, 6 | lb2d 58 | . 2 ⊦ (~ 𝜑 ⅋ (𝜓 ⧟ 𝜒)) |
8 | 7 | dfli2i 66 | 1 ⊦ (𝜑 ⊸ (𝜓 ⧟ 𝜒)) |
Colors of variables: wff var nilad |
Syntax hints: ~ wneg 3 & wac 30 ⧟ wlb 55 ⊸ wli 61 |
This theorem was proved from axioms: ax-ibot 4 ax-ebot 5 ax-cut 6 ax-init 7 ax-mdco 8 ax-mdas 9 ax-iac 32 ax-eac1 33 ax-eac2 34 |
This theorem depends on definitions: df-lb 56 df-li 62 |
This theorem is referenced by: lbsymd 102 |
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