Theorem ilb 96

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

Hypotheses

Ref	Expression
ilb.1	⊦ (𝜑 ⊸ 𝜓)
ilb.2	⊦ (𝜓 ⊸ 𝜑)

Assertion

Ref	Expression
ilb	⊦ (𝜑 ⧟ 𝜓)

Proof of Theorem ilb

Step	Hyp	Ref	Expression
1		ilb.1	. . 3 ⊦ (𝜑 ⊸ 𝜓)
2		ilb.2	. . 3 ⊦ (𝜓 ⊸ 𝜑)
3	1, 2	iaci 36	. 2 ⊦ ((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑))
4		dflb 93	. 2 ⊦ ((𝜑 ⧟ 𝜓) ⧟ ((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑)))
5	3, 4	lb2i 60	1 ⊦ (𝜑 ⧟ 𝜓)

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:  lbrf  98  lbeui  99  dn  107  mdcob  109  md1  113  md2  114  mccob  116