Theorem anxordi 1520

Description: Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 1013 does not necessarily hold, in part because the usual definition of xor is subtly different in intuitionistic logic. (Contributed by David A. Wheeler, 7-Oct-2018.)

Assertion

Ref	Expression
anxordi	⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)))

Proof of Theorem anxordi

Step	Hyp	Ref	Expression
1		xordi 1013	. 2 ⊢ ((𝜑 ∧ ¬ (𝜓 ↔ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
2		df-xor 1504	. . 3 ⊢ ((𝜓 ⊻ 𝜒) ↔ ¬ (𝜓 ↔ 𝜒))
3	2	anbi2i 622	. 2 ⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ∧ ¬ (𝜓 ↔ 𝜒)))
4		df-xor 1504	. 2 ⊢ (((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
5	1, 3, 4	3bitr4i 302	1 ⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ⊻ wxo 1503

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 206  df-an 396  df-xor 1504

This theorem is referenced by: (None)