Theorem ilbd 97

Description: Construct a biconditional from its forward and reverse implications.

Hypotheses

Ref	Expression
ilbd.1	⊦ (𝜑 ⊸ (𝜓 ⊸ 𝜒))
ilbd.2	⊦ (𝜑 ⊸ (𝜒 ⊸ 𝜓))

Assertion

Ref	Expression
ilbd	⊦ (𝜑 ⊸ (𝜓 ⧟ 𝜒))

Proof of Theorem ilbd

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))

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