Theorem birev-P2.5b 115

Description: '↔' Reverse Implication.

Assertion

Ref	Expression
birev-P2.5b	⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))

Proof of Theorem birev-P2.5b

Step	Hyp	Ref	Expression
1		dfbionlyif-P2.3b 109	. 2 ⊢ ((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
2		simpr-L2.2b 97	. 2 ⊢ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜓 → 𝜑))
3	1, 2	syl-P1.2 34	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))

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:  birev-P2.5b.SH  116  birev-P2.5b.AC.SH  117  birev-P2.5b.2AC.SH  118  bisym-P2.6b  124  subneg-P2.7  127  subiml-P2.8a  128  subimr-P2.8b  130