Theorem bicmb-P2.5c 119

Description: '↔' Combine Implications.

Assertion

Ref	Expression
bicmb-P2.5c	⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))

Proof of Theorem bicmb-P2.5c

Step	Hyp	Ref	Expression
1		cmb-L2.3 99	. 2 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))))
2		dfbiif-P2.3a 108	. . . 4 ⊢ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))
3	2	axL1.SH 30	. . 3 ⊢ ((𝜓 → 𝜑) → (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
4	3	axL1.SH 30	. 2 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))))
5	1, 4	mpt-P1.8.2AC.2SH 59	1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107

This theorem is referenced by:  bicmb-P2.5c.2SH  120  bicmb-P2.5c.AC.2SH  121  bicmb-P2.5c.2AC.2SH  122