Theorem bifwd-P2.5a 111

Description: '↔' Forward Implication.

Assertion

Ref	Expression
bifwd-P2.5a	⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))

Proof of Theorem bifwd-P2.5a

Step	Hyp	Ref	Expression
1		dfbionlyif-P2.3b 109	. 2 ⊢ ((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
2		simpl-L2.2a 95	. 2 ⊢ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 → 𝜓))
3	1, 2	syl-P1.2 34	1 ⊢ ((𝜑 ↔ 𝜓) → (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:  bifwd-P2.5a.SH  112  bifwd-P2.5a.AC.SH  113  bifwd-P2.5a.2AC.SH  114  bisym-P2.6b  124  subneg-P2.7  127  subiml-P2.8a  128  subimr-P2.8b  130