Theorem xorbi12i

Description: Equality property for exclusive disjunction. (Contributed by Mario Carneiro, 4-Sep-2016) (Proof shortened by Wolf Lammen, 21-Apr-2024)

	Ref	Expression
Hypotheses	xorbi12.1	⊢ ( 𝜑 ↔ 𝜓 )
	xorbi12.2	⊢ ( 𝜒 ↔ 𝜃 )
Assertion	xorbi12i	⊢ ( ( 𝜑 ⊻ 𝜒 ) ↔ ( 𝜓 ⊻ 𝜃 ) )

Proof

Step	Hyp	Ref	Expression
1		xorbi12.1	⊢ ( 𝜑 ↔ 𝜓 )
2		xorbi12.2	⊢ ( 𝜒 ↔ 𝜃 )
3		df-xor	⊢ ( ( 𝜑 ⊻ 𝜒 ) ↔ ¬ ( 𝜑 ↔ 𝜒 ) )
4	1 2	bibi12i	⊢ ( ( 𝜑 ↔ 𝜒 ) ↔ ( 𝜓 ↔ 𝜃 ) )
5	3 4	xchbinx	⊢ ( ( 𝜑 ⊻ 𝜒 ) ↔ ¬ ( 𝜓 ↔ 𝜃 ) )
6		df-xor	⊢ ( ( 𝜓 ⊻ 𝜃 ) ↔ ¬ ( 𝜓 ↔ 𝜃 ) )
7	5 6	bitr4i	⊢ ( ( 𝜑 ⊻ 𝜒 ) ↔ ( 𝜓 ⊻ 𝜃 ) )