Theorem prlem2 1071

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.)

Assertion

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

Proof of Theorem prlem2

Step	Hyp	Ref	Expression
1		simpl 488	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2		simpl 488	. . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜒)
3	1, 2	orim12i 922	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) → (𝜑 ∨ 𝜒))
4	3	pm4.71ri 570	1 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∧ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ wo 861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862

This theorem is used by:  zfpair  5394