Theorem dflb1s 91

Description: Forward implication of biconditional definition.

Assertion

Ref	Expression
dflb1s	⊦ ((𝜑 ⧟ 𝜓) ⊸ ((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑)))

Proof of Theorem dflb1s

Step	Hyp	Ref	Expression
1		lbi1s 87	. . . 4 ⊦ ((𝜑 ⧟ 𝜓) ⊸ (𝜑 ⊸ 𝜓))
2	1	dfli1i 65	. . 3 ⊦ (~ (𝜑 ⧟ 𝜓) ⅋ (𝜑 ⊸ 𝜓))
3		lbi2s 88	. . . 4 ⊦ ((𝜑 ⧟ 𝜓) ⊸ (𝜓 ⊸ 𝜑))
4	3	dfli1i 65	. . 3 ⊦ (~ (𝜑 ⧟ 𝜓) ⅋ (𝜓 ⊸ 𝜑))
5	2, 4	ax-iac 32	. 2 ⊦ (~ (𝜑 ⧟ 𝜓) ⅋ ((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑)))
6	5	dfli2i 66	1 ⊦ ((𝜑 ⧟ 𝜓) ⊸ ((𝜑 ⊸ 𝜓) & (𝜓 ⊸ 𝜑)))

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-iac 32  ax-eac1 33  ax-eac2 34

This theorem depends on definitions:  df-lb 56  df-li 62

This theorem is referenced by:  dflb  93