Theorem anxordi 1556

Description: Conjunction distributes over exclusive-or. In intuitionistic logic this assertion is also true, even though xordi 1034 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 1034	. 2 ⊢ ((𝜑 ∧ ¬ (𝜓 ↔ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
2		df-xor 1542	. . 3 ⊢ ((𝜓 ⊻ 𝜒) ↔ ¬ (𝜓 ↔ 𝜒))
3	2	anbi2i 634	. 2 ⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ (𝜑 ∧ ¬ (𝜓 ↔ 𝜒)))
4		df-xor 1542	. 2 ⊢ (((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)) ↔ ¬ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜒)))
5	1, 3, 4	3bitr4i 306	1 ⊢ ((𝜑 ∧ (𝜓 ⊻ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ⊻ (𝜑 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ⊻ wxo 1541

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 210  df-an 401  df-xor 1542

This theorem is referenced by: (None)