Theorem xornbi 1277

Description: A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1282. (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 1275	. 2 ⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ↔ ¬ 𝜓))
2		pm5.18im 1276	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))
3	2	con2i 557	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) → ¬ (𝜑 ↔ 𝜓))
4	1, 3	syl 14	1 ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ⊻ wxo 1266

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  ax-io 630

This theorem depends on definitions:  df-bi 110  df-xor 1267

This theorem is referenced by: (None)