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Theorem prlem2 1039
 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 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))

Proof of Theorem prlem2
StepHypRef Expression
1 simpl 476 . . 3 ((𝜑𝜓) → 𝜑)
2 simpl 476 . . 3 ((𝜒𝜃) → 𝜒)
31, 2orim12i 895 . 2 (((𝜑𝜓) ∨ (𝜒𝜃)) → (𝜑𝜒))
43pm4.71ri 556 1 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ wa 386   ∨ wo 836 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 199  df-an 387  df-or 837 This theorem is referenced by:  zfpair  5136
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