Theorem xorbi12i 1515

Description: Equality property for XOR. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 21-Apr-2024.)

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		df-xor 1502	. . 3 ⊢ ((𝜑 ⊻ 𝜒) ↔ ¬ (𝜑 ↔ 𝜒))
2		xorbi12.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3		xorbi12.2	. . . 4 ⊢ (𝜒 ↔ 𝜃)
4	2, 3	bibi12i 342	. . 3 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃))
5	1, 4	xchbinx 336	. 2 ⊢ ((𝜑 ⊻ 𝜒) ↔ ¬ (𝜓 ↔ 𝜃))
6		df-xor 1502	. 2 ⊢ ((𝜓 ⊻ 𝜃) ↔ ¬ (𝜓 ↔ 𝜃))
7	5, 6	bitr4i 280	1 ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))

Syntax hints:  ¬ wn 3   ↔ wb 208   ⊻ wxo 1501

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 1502

This theorem is referenced by:  hadcomaOLD  1600  hadcomb  1601  symdifass  4230