Theorem dfbiif-P2.3a 108

Description: Sufficient Conditon for (i.e. "If" part of) df-bi-D2.1 107.

Assertion

Ref	Expression
dfbiif-P2.3a	⊢ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓))

Proof of Theorem dfbiif-P2.3a

Step	Hyp	Ref	Expression
1		df-bi-D2.1 107	. 2 ⊢ ¬ (((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑))) → ¬ (¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
2	1	simpr-L2.2b.SH 98	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:  dfbialt-P2.4  110  bicmb-P2.5c  119