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Theorem prlem2 941
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 108 . . 3 ((𝜑𝜓) → 𝜑)
2 simpl 108 . . 3 ((𝜒𝜃) → 𝜒)
31, 2orim12i 731 . 2 (((𝜑𝜓) ∨ (𝜒𝜃)) → (𝜑𝜒))
43pm4.71ri 387 1 (((𝜑𝜓) ∨ (𝜒𝜃)) ↔ ((𝜑𝜒) ∧ ((𝜑𝜓) ∨ (𝜒𝜃))))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wo 680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
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