Theorem prlem2 929

Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 5-Aug-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 443	. . 3 ⊢ ((φ ∧ ψ) → φ)
2		simpl 443	. . 3 ⊢ ((χ ∧ θ) → χ)
3	1, 2	orim12i 502	. 2 ⊢ (((φ ∧ ψ) ∨ (χ ∧ θ)) → (φ ∨ χ))
4	3	pm4.71ri 614	1 ⊢ (((φ ∧ ψ) ∨ (χ ∧ θ)) ↔ ((φ ∨ χ) ∧ ((φ ∧ ψ) ∨ (χ ∧ θ))))

Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358

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 177  df-or 359  df-an 360

This theorem is referenced by: (None)