Theorem xornbi 1364

Description: A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1369. (Contributed by Jim Kingdon, 10-Mar-2018.)

Assertion

Ref	Expression
xornbi	⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ↔ 𝜓))

Proof of Theorem xornbi

Step	Hyp	Ref	Expression
1		xorbin 1362	. 2 ⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ↔ ¬ 𝜓))
2		pm5.18im 1363	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))
3	2	con2i 616	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) → ¬ (𝜑 ↔ 𝜓))
4	1, 3	syl 14	1 ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ⊻ wxo 1353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-xor 1354

This theorem is referenced by: (None)