Theorem xorbi12i 1274

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	1	a1i 9	. . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓))
3		xorbi12.2	. . . 4 ⊢ (𝜒 ↔ 𝜃)
4	3	a1i 9	. . 3 ⊢ (⊤ → (𝜒 ↔ 𝜃))
5	2, 4	xorbi12d 1273	. 2 ⊢ (⊤ → ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃)))
6	5	trud 1252	1 ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))

Syntax hints:   ↔ wb 98  ⊤wtru 1244   ⊻ wxo 1266

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-xor 1267

This theorem is referenced by: (None)