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Theorem prlem1 915
Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 5-Jan-2013.)
Hypotheses
Ref Expression
prlem1.1 (𝜑 → (𝜂𝜒))
prlem1.2 (𝜓 → ¬ 𝜃)
Assertion
Ref Expression
prlem1 (𝜑 → (𝜓 → (((𝜓𝜒) ∨ (𝜃𝜏)) → 𝜂)))

Proof of Theorem prlem1
StepHypRef Expression
1 prlem1.1 . . . . 5 (𝜑 → (𝜂𝜒))
21biimprd 156 . . . 4 (𝜑 → (𝜒𝜂))
32adantld 272 . . 3 (𝜑 → ((𝜓𝜒) → 𝜂))
4 prlem1.2 . . . . 5 (𝜓 → ¬ 𝜃)
54pm2.21d 582 . . . 4 (𝜓 → (𝜃𝜂))
65adantrd 273 . . 3 (𝜓 → ((𝜃𝜏) → 𝜂))
73, 6jaao 672 . 2 ((𝜑𝜓) → (((𝜓𝜒) ∨ (𝜃𝜏)) → 𝜂))
87ex 113 1 (𝜑 → (𝜓 → (((𝜓𝜒) ∨ (𝜃𝜏)) → 𝜂)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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