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Metamath Proof Explorer

Theorem xordi

Description: Conjunction distributes over exclusive-or, using -. ( ph <-> ps ) to express exclusive-or. This is one way to interpret the distributive law of multiplication over addition in modulo 2 arithmetic. This is not necessarily true in intuitionistic logic, though anxordi does hold in it. (Contributed by NM, 3-Oct-2008)

		Ref	Expression
	Assertion	xordi	⊢ ( ( 𝜑 ∧ ¬ ( 𝜓 ↔ 𝜒 ) ) ↔ ¬ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		annim	⊢ ( ( 𝜑 ∧ ¬ ( 𝜓 ↔ 𝜒 ) ) ↔ ¬ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) )
2		pm5.32	⊢ ( ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜒 ) ) )
3	1 2	xchbinx	⊢ ( ( 𝜑 ∧ ¬ ( 𝜓 ↔ 𝜒 ) ) ↔ ¬ ( ( 𝜑 ∧ 𝜓 ) ↔ ( 𝜑 ∧ 𝜒 ) ) )