Theorem xnor 1513

Description: Two ways to write XNOR (exclusive not-or). (Contributed by Mario Carneiro, 4-Sep-2016.)

Assertion

Ref	Expression
xnor	⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))

Proof of Theorem xnor

Step	Hyp	Ref	Expression
1		df-xor 1512	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
2	1	con2bii 357	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (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:  xorass  1515  xorneg2  1521  hadbi  1598  had0  1604  wl-df-3xor  37469  wl-3xorbi  37474  tsxo1  38144  tsxo2  38145