Theorem lbsymd 102

Description: Linear biconditional is symmetric. Deduction for lbsym 127.

Hypothesis

Ref	Expression
lbsymd.1	⊦ (𝜑 ⊸ (𝜓 ⧟ 𝜒))

Assertion

Ref	Expression
lbsymd	⊦ (𝜑 ⊸ (𝜒 ⧟ 𝜓))

Proof of Theorem lbsymd

Step	Hyp	Ref	Expression
1		lbsymd.1	. . . . . 6 ⊦ (𝜑 ⊸ (𝜓 ⧟ 𝜒))
2	1	dfli1i 65	. . . . 5 ⊦ (~ 𝜑 ⅋ (𝜓 ⧟ 𝜒))
3		dflb 93	. . . . 5 ⊦ ((𝜓 ⧟ 𝜒) ⧟ ((𝜓 ⊸ 𝜒) & (𝜒 ⊸ 𝜓)))
4	2, 3	lb1d 57	. . . 4 ⊦ (~ 𝜑 ⅋ ((𝜓 ⊸ 𝜒) & (𝜒 ⊸ 𝜓)))
5	4	ax-eac2 34	. . 3 ⊦ (~ 𝜑 ⅋ (𝜒 ⊸ 𝜓))
6	5	dfli2i 66	. 2 ⊦ (𝜑 ⊸ (𝜒 ⊸ 𝜓))
7	4	ax-eac1 33	. . 3 ⊦ (~ 𝜑 ⅋ (𝜓 ⊸ 𝜒))
8	7	dfli2i 66	. 2 ⊦ (𝜑 ⊸ (𝜓 ⊸ 𝜒))
9	6, 8	ilbd 97	1 ⊦ (𝜑 ⊸ (𝜒 ⧟ 𝜓))

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)