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Theorem lbsymd 102
Description: Linear biconditional is symmetric. Deduction for lbsym 127.
Hypothesis
Ref Expression
lbsymd.1 (𝜑 ⊸ (𝜓𝜒))
Assertion
Ref Expression
lbsymd (𝜑 ⊸ (𝜒𝜓))

Proof of Theorem lbsymd
StepHypRef Expression
1 lbsymd.1 . . . . . 6 (𝜑 ⊸ (𝜓𝜒))
21dfli1i 65 . . . . 5 (~ 𝜑 ⅋ (𝜓𝜒))
3 dflb 93 . . . . 5 ((𝜓𝜒) ⧟ ((𝜓𝜒) & (𝜒𝜓)))
42, 3lb1d 57 . . . 4 (~ 𝜑 ⅋ ((𝜓𝜒) & (𝜒𝜓)))
54ax-eac2 34 . . 3 (~ 𝜑 ⅋ (𝜒𝜓))
65dfli2i 66 . 2 (𝜑 ⊸ (𝜒𝜓))
74ax-eac1 33 . . 3 (~ 𝜑 ⅋ (𝜓𝜒))
87dfli2i 66 . 2 (𝜑 ⊸ (𝜓𝜒))
96, 8ilbd 97 1 (𝜑 ⊸ (𝜒𝜓))
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: (None)
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