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Theorem prlem2 892
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 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))

Proof of Theorem prlem2
StepHypRef Expression
1 simpl 106 . . 3 ((𝜑𝜓) → 𝜑)
2 simpl 106 . . 3 ((𝜒𝜃) → 𝜒)
31, 2orim12i 686 . 2 (((𝜑𝜓) ∨ (𝜒𝜃)) → (𝜑𝜒))
43pm4.71ri 378 1 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wo 639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640
This theorem depends on definitions:  df-bi 114
This theorem is referenced by: (None)
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