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Theorem oplem1 930
 Description: A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.)
Hypotheses
Ref Expression
oplem1.1 (φ → (ψ χ))
oplem1.2 (φ → (θ τ))
oplem1.3 (ψθ)
oplem1.4 (χ → (θτ))
Assertion
Ref Expression
oplem1 (φψ)

Proof of Theorem oplem1
StepHypRef Expression
1 oplem1.3 . . . . . . 7 (ψθ)
21notbii 287 . . . . . 6 ψ ↔ ¬ θ)
3 oplem1.1 . . . . . . 7 (φ → (ψ χ))
43ord 366 . . . . . 6 (φ → (¬ ψχ))
52, 4syl5bir 209 . . . . 5 (φ → (¬ θχ))
6 oplem1.2 . . . . . 6 (φ → (θ τ))
76ord 366 . . . . 5 (φ → (¬ θτ))
85, 7jcad 519 . . . 4 (φ → (¬ θ → (χ τ)))
9 oplem1.4 . . . . 5 (χ → (θτ))
109biimpar 471 . . . 4 ((χ τ) → θ)
118, 10syl6 29 . . 3 (φ → (¬ θθ))
1211pm2.18d 103 . 2 (φθ)
1312, 1sylibr 203 1 (φψ)
 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-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360 This theorem is referenced by:  preqr1  4124
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