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Theorem oplem1 870
Description: A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
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 oplem1.1 . 2 (φ → (ψ χ))
2 idd 21 . . 3 (φ → (ψψ))
3 oplem1.2 . . . . 5 (φ → (θ τ))
4 ax-1 5 . . . . . 6 (θ → (χθ))
5 oplem1.4 . . . . . . 7 (χ → (θτ))
65biimprcd 149 . . . . . 6 (τ → (χθ))
74, 6jaoi 623 . . . . 5 ((θ τ) → (χθ))
83, 7syl 14 . . . 4 (φ → (χθ))
9 oplem1.3 . . . 4 (ψθ)
108, 9syl6ibr 151 . . 3 (φ → (χψ))
112, 10jaod 624 . 2 (φ → ((ψ χ) → ψ))
121, 11mpd 13 1 (φψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   wo 616
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-io 617
This theorem depends on definitions:  df-bi 110
This theorem is referenced by:  preqr1g  3511  preqr1  3513
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