Theorem bj-bixor

Description: Equivalence of two ternary operations. Note the identical order and parenthesizing of the three arguments in both expressions. (Contributed by BJ, 31-Dec-2023)

	Ref	Expression
Assertion	bj-bixor	⊢ ( ( 𝜑 ↔ ( 𝜓 ⊻ 𝜒 ) ) ↔ ( 𝜑 ⊻ ( 𝜓 ↔ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.18	⊢ ( ( 𝜑 ↔ ( 𝜓 ↔ 𝜒 ) ) ↔ ¬ ( 𝜑 ↔ ¬ ( 𝜓 ↔ 𝜒 ) ) )
2	1	con2bii	⊢ ( ( 𝜑 ↔ ¬ ( 𝜓 ↔ 𝜒 ) ) ↔ ¬ ( 𝜑 ↔ ( 𝜓 ↔ 𝜒 ) ) )
3		df-xor	⊢ ( ( 𝜓 ⊻ 𝜒 ) ↔ ¬ ( 𝜓 ↔ 𝜒 ) )
4	3	bibi2i	⊢ ( ( 𝜑 ↔ ( 𝜓 ⊻ 𝜒 ) ) ↔ ( 𝜑 ↔ ¬ ( 𝜓 ↔ 𝜒 ) ) )
5		df-xor	⊢ ( ( 𝜑 ⊻ ( 𝜓 ↔ 𝜒 ) ) ↔ ¬ ( 𝜑 ↔ ( 𝜓 ↔ 𝜒 ) ) )
6	2 4 5	3bitr4i	⊢ ( ( 𝜑 ↔ ( 𝜓 ⊻ 𝜒 ) ) ↔ ( 𝜑 ⊻ ( 𝜓 ↔ 𝜒 ) ) )