Theorem pm5.18im 1396

Description: One direction of pm5.18dc 884, 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		pm5.19 707	. 2 ⊢ ¬ (𝜓 ↔ ¬ 𝜓)
2		bibi1 240	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ ¬ 𝜓) ↔ (𝜓 ↔ ¬ 𝜓)))
3	2	notbid 668	. 2 ⊢ ((𝜑 ↔ 𝜓) → (¬ (𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜓 ↔ ¬ 𝜓)))
4	1, 3	mpbiri 168	1 ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  xornbi  1397