Theorem dfbionlyif-P2.3b 109

Description: Necessary Condition for (i.e. "Only If" part of) df-bi-D2.1 107.

Assertion

Ref	Expression
dfbionlyif-P2.3b	⊢ ((𝜑 ↔ 𝜓) → ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))

Proof of Theorem dfbionlyif-P2.3b

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	simpl-L2.2a.SH 96	1 ⊢ ((𝜑 ↔ 𝜓) → ¬ ((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:  bifwd-P2.5a  111  birev-P2.5b  115