Theorem bj-bixor 33925

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.)

Assertion

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

Proof of Theorem bj-bixor

Step	Hyp	Ref	Expression
1		pm5.18 385	. . 3 ⊢ ((𝜑 ↔ (𝜓 ↔ 𝜒)) ↔ ¬ (𝜑 ↔ ¬ (𝜓 ↔ 𝜒)))
2	1	con2bii 360	. 2 ⊢ ((𝜑 ↔ ¬ (𝜓 ↔ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ↔ 𝜒)))
3		df-xor 1502	. . 3 ⊢ ((𝜓 ⊻ 𝜒) ↔ ¬ (𝜓 ↔ 𝜒))
4	3	bibi2i 340	. 2 ⊢ ((𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ↔ ¬ (𝜓 ↔ 𝜒)))
5		df-xor 1502	. 2 ⊢ ((𝜑 ⊻ (𝜓 ↔ 𝜒)) ↔ ¬ (𝜑 ↔ (𝜓 ↔ 𝜒)))
6	2, 4, 5	3bitr4i 305	1 ⊢ ((𝜑 ↔ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ⊻ (𝜓 ↔ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 208   ⊻ wxo 1501

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 209  df-xor 1502

This theorem is referenced by: (None)