xor3 - Metamath Proof Explorer

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Theorem xor3 382

Description: Two ways to express "exclusive or". (Contributed by NM, 1-Jan-2006.)

**Assertion**
Ref	Expression
xor3	⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

**Proof of Theorem xor3**
Step	Hyp	Ref	Expression
1		pm5.18 381	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓))
2	1	con2bii 357	. 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ ¬ (𝜑 ↔ 𝜓))
3	2	bicomi 223	1 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ (𝜑 ↔ ¬ 𝜓))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 205

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 206

This theorem is referenced by: nbbn 383 pm5.15 1011 nbi2 1014 xorass 1509 hadnot 1596 nabbib 3042 nmogtmnf 30593 nmopgtmnf 31691 limsucncmpi 35929 wl-3xorbi 36952 wl-3xornot 36960 oneptri 42685 oaordnrex 42724 omnord1ex 42733 oenord1ex 42744 aiffnbandciffatnotciffb 46286 axorbciffatcxorb 46287 abnotbtaxb 46297 afv2orxorb 46608 line2ylem 47824 line2xlem 47826

	
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