Theorem axorbtnotaiffb 46899

Description: Given a is exclusive to b, there exists a proof for (not (a if-and-only-if b)); df-xor 1512 is a closed form of this. (Contributed by Jarvin Udandy, 7-Sep-2016.)

Hypothesis

Ref	Expression
axorbtnotaiffb.1	⊢ (𝜑 ⊻ 𝜓)

Assertion

Ref	Expression
axorbtnotaiffb	⊢ ¬ (𝜑 ↔ 𝜓)

Proof of Theorem axorbtnotaiffb

Step	Hyp	Ref	Expression
1		axorbtnotaiffb.1	. 2 ⊢ (𝜑 ⊻ 𝜓)
2		df-xor 1512	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3	1, 2	mpbi 230	1 ⊢ ¬ (𝜑 ↔ 𝜓)

Syntax hints:  ¬ wn 3   ↔ wb 206   ⊻ wxo 1511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-xor 1512

This theorem is referenced by:  axorbciffatcxorb  46901  aifftbifffaibifff  46918