Theorem prlem2

Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 21-Jun-1993) (Proof shortened by Andrew Salmon, 13-May-2011) (Proof shortened by Wolf Lammen, 9-Dec-2012)

	Ref	Expression
Assertion	prlem2	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∨ 𝜒 ) ∧ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ 𝜃 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜑 )
2		simpl	⊢ ( ( 𝜒 ∧ 𝜃 ) → 𝜒 )
3	1 2	orim12i	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ 𝜃 ) ) → ( 𝜑 ∨ 𝜒 ) )
4	3	pm4.71ri	⊢ ( ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ 𝜃 ) ) ↔ ( ( 𝜑 ∨ 𝜒 ) ∧ ( ( 𝜑 ∧ 𝜓 ) ∨ ( 𝜒 ∧ 𝜃 ) ) ) )