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Theorem prlem1 1050
 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 251 . . . 4 (𝜑 → (𝜒𝜂))
32adantld 494 . . 3 (𝜑 → ((𝜓𝜒) → 𝜂))
4 prlem1.2 . . . . 5 (𝜓 → ¬ 𝜃)
54pm2.21d 121 . . . 4 (𝜓 → (𝜃𝜂))
65adantrd 495 . . 3 (𝜓 → ((𝜃𝜏) → 𝜂))
73, 6jaao 952 . 2 ((𝜑𝜓) → (((𝜓𝜒) ∨ (𝜃𝜏)) → 𝜂))
87ex 416 1 (𝜑 → (𝜓 → (((𝜓𝜒) ∨ (𝜃𝜏)) → 𝜂)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844 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 210  df-an 400  df-or 845 This theorem is referenced by:  zfpair  5290
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