Theorem pm5.18im 1276

Description: One direction of pm5.18dc 777, which holds for all propositions, not just decidable propositions. (Contributed by Jim Kingdon, 10-Mar-2018.)

Assertion

Ref	Expression
pm5.18im	⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))

Proof of Theorem pm5.18im

Step	Hyp	Ref	Expression
1		pm5.19 622	. 2 ⊢ ¬ (𝜓 ↔ ¬ 𝜓)
2		bibi1 229	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜓)))
3	2	notbid 592	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜓 ↔ ¬ 𝜓)))
4	1, 3	mpbiri 157	1 ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  xornbi  1277