Theorem xorbi12d

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

	Ref	Expression
Hypotheses	xor12d.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
	xor12d.2	⊢ ( 𝜑 → ( 𝜃 ↔ 𝜏 ) )
Assertion	xorbi12d	⊢ ( 𝜑 → ( ( 𝜓 ⊻ 𝜃 ) ↔ ( 𝜒 ⊻ 𝜏 ) ) )

Proof

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