Theorem xorbi12i 1361

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 1360	. 2 ⊢ (⊤ → ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃)))
6	5	mptru 1340	1 ⊢ ((𝜑 ⊻ 𝜒) ↔ (𝜓 ⊻ 𝜃))

Syntax hints:   ↔ wb 104  ⊤wtru 1332   ⊻ wxo 1353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-xor 1354

This theorem is referenced by: (None)