Theorem prlem2 998

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 472	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
2		simpl 472	. . 3 ⊢ ((𝜒 ∧ 𝜃) → 𝜒)
3	1, 2	orim12i 537	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) → (𝜑 ∨ 𝜒))
4	3	pm4.71ri 663	1 ⊢ (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∨ 𝜒) ∧ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ 𝜃))))

Syntax hints:   ↔ wb 195   ∨ wo 382   ∧ wa 383

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 196  df-or 384  df-an 385

This theorem is referenced by:  zfpair  4831