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Mirrors > Home > LLPE Home > Th. List > dflb | Structured version |
Description: Nicer definition of biconditional. Uses ⧟ and ⊸. |
Ref | Expression |
---|---|
dflb | ⊦ ((𝜑 ⧟ 𝜓) ⧟ ((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dflb1s 91 | . . 3 ⊦ ((𝜑 ⧟ 𝜓) ⊸ ((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑))) | |
2 | dflb2s 92 | . . 3 ⊦ (((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑)) ⊸ (𝜑 ⧟ 𝜓)) | |
3 | 1, 2 | iaci 36 | . 2 ⊦ (((𝜑 ⧟ 𝜓) ⊸ ((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑))) & (((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑)) ⊸ (𝜑 ⧟ 𝜓))) |
4 | dflb2s 92 | . 2 ⊦ ((((𝜑 ⧟ 𝜓) ⊸ ((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑))) & (((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑)) ⊸ (𝜑 ⧟ 𝜓))) ⊸ ((𝜑 ⧟ 𝜓) ⧟ ((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑)))) | |
5 | 3, 4 | lmp 76 | 1 ⊦ ((𝜑 ⧟ 𝜓) ⧟ ((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑))) |
Colors of variables: wff var nilad |
Syntax hints: & 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: ilb 96 ilbd 97 lbsymd 102 abs1 178 |
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