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Theorem prlem1 928
 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 → (((ψ χ) (θ τ)) → η)))

Proof of Theorem prlem1
StepHypRef Expression
1 prlem1.1 . . . . 5 (φ → (ηχ))
21biimprd 214 . . . 4 (φ → (χη))
32adantld 453 . . 3 (φ → ((ψ χ) → η))
4 prlem1.2 . . . . 5 (ψ → ¬ θ)
54pm2.21d 98 . . . 4 (ψ → (θη))
65adantrd 454 . . 3 (ψ → ((θ τ) → η))
73, 6jaao 495 . 2 ((φ ψ) → (((ψ χ) (θ τ)) → η))
87ex 423 1 (φ → (ψ → (((ψ χ) (θ τ)) → η)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360 This theorem is referenced by: (None)
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