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Theorem prlem1 880
 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 147 . . . 4 (𝜑 → (𝜒𝜂))
32adantld 263 . . 3 (𝜑 → ((𝜓𝜒) → 𝜂))
4 prlem1.2 . . . . 5 (𝜓 → ¬ 𝜃)
54pm2.21d 549 . . . 4 (𝜓 → (𝜃𝜂))
65adantrd 264 . . 3 (𝜓 → ((𝜃𝜏) → 𝜂))
73, 6jaao 639 . 2 ((𝜑𝜓) → (((𝜓𝜒) ∨ (𝜃𝜏)) → 𝜂))
87ex 108 1 (𝜑 → (𝜓 → (((𝜓𝜒) ∨ (𝜃𝜏)) → 𝜂)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110 This theorem is referenced by: (None)
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