Theorem anxordi 1518

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 1502	. . 3 ⊢ ((𝜓 ⊻ 𝜒) ↔ ¬ (𝜓 ↔ 𝜒))
3	2	anbi2i 624	. 2 ⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ∧ ¬ (𝜓 ↔ 𝜒)))
4		df-xor 1502	. 2 ⊢ (((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
5	1, 3, 4	3bitr4i 305	1 ⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ⊻ 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-an 399  df-xor 1502

This theorem is referenced by: (None)