Theorem xorbi12d 1525

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

Hypotheses

Ref	Expression
xor12d.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
xor12d.2	⊢ (𝜑 → (𝜃 ↔ 𝜏))

Assertion

Ref	Expression
xorbi12d	⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏)))

Proof of Theorem xorbi12d

Step	Hyp	Ref	Expression
1		xor12d.1	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		xor12d.2	. . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏))
3	1, 2	bibi12d 345	. . 3 ⊢ (𝜑 → ((𝜓 ↔ 𝜃) ↔ (𝜒 ↔ 𝜏)))
4	3	notbid 318	. 2 ⊢ (𝜑 → (¬ (𝜓 ↔ 𝜃) ↔ ¬ (𝜒 ↔ 𝜏)))
5		df-xor 1512	. 2 ⊢ ((𝜓 ⊻ 𝜃) ↔ ¬ (𝜓 ↔ 𝜃))
6		df-xor 1512	. 2 ⊢ ((𝜒 ⊻ 𝜏) ↔ ¬ (𝜒 ↔ 𝜏))
7	4, 5, 6	3bitr4g 314	1 ⊢ (𝜑 → ((𝜓 ⊻ 𝜃) ↔ (𝜒 ⊻ 𝜏)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ⊻ wxo 1511

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 207  df-xor 1512

This theorem is referenced by:  hadbi123d  1595  cadbi123d  1610