Theorem xnor 1542

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 1541	. 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
2	1	con2bii 360	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ⊻ 𝜓))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ⊻ wxo 1540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-xor 1541

This theorem is used by:  xorass  1544  xorneg2  1550  hadbi  1627  had0OLD  1635  wl-df-3xor  38392  wl-3xorbi  38397  tsxo1  39070  tsxo2  39071