Theorem xorbi12i 1513

Description: Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.)

Hypotheses

Ref	Expression
xorbi12.1	⊢ (𝜑 ↔ 𝜓)
xorbi12.2	⊢ (𝜒 ↔ 𝜃)

Assertion

Ref	Expression
xorbi12i	⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))

Proof of Theorem xorbi12i

Step	Hyp	Ref	Expression
1		xorbi12.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2		xorbi12.2	. . . 4 ⊢ (𝜒 ↔ 𝜃)
3	1, 2	bibi12i 342	. . 3 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃))
4	3	notbii 322	. 2 ⊢ (¬ (𝜑 ↔ 𝜒) ↔ ¬ (𝜓 ↔ 𝜃))
5		df-xor 1501	. 2 ⊢ ((𝜑 ⊻ 𝜒) ↔ ¬ (𝜑 ↔ 𝜒))
6		df-xor 1501	. 2 ⊢ ((𝜓 ⊻ 𝜃) ↔ ¬ (𝜓 ↔ 𝜃))
7	4, 5, 6	3bitr4i 305	1 ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))

Syntax hints:  ¬ wn 3   ↔ wb 208   ⊻ wxo 1500

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 209  df-xor 1501

This theorem is referenced by:  hadcomaOLD  1596  hadcomb  1597  symdifass  4227