Theorem xorbi12i 1314

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 1313	. 2 ⊢ (⊤ → ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃)))
6	5	trud 1293	1 ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))

Syntax hints:   ↔ wb 103  ⊤wtru 1285   ⊻ wxo 1306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-xor 1307

This theorem is referenced by: (None)