Theorem xornbi 1397

Description: A consequence of exclusive or. For decidable propositions this is an equivalence, as seen at xornbidc 1402. (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 1395	. 2 ⊢ ((𝜑 ⊻ 𝜓) → (𝜑 ↔ ¬ 𝜓))
2		pm5.18im 1396	. . 3 ⊢ ((𝜑 ↔ 𝜓) → ¬ (𝜑 ↔ ¬ 𝜓))
3	2	con2i 628	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) → ¬ (𝜑 ↔ 𝜓))
4	1, 3	syl 14	1 ⊢ ((𝜑 ⊻ 𝜓) → ¬ (𝜑 ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ⊻ wxo 1386

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  ax-io 710

This theorem depends on definitions:  df-bi 117  df-xor 1387

This theorem is referenced by: (None)