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Theorem oplem1 891
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 → (𝜓𝜒))
oplem1.2 (𝜑 → (𝜃𝜏))
oplem1.3 (𝜓𝜃)
oplem1.4 (𝜒 → (𝜃𝜏))
Assertion
Ref Expression
oplem1 (𝜑𝜓)

Proof of Theorem oplem1
StepHypRef Expression
1 oplem1.1 . 2 (𝜑 → (𝜓𝜒))
2 idd 21 . . 3 (𝜑 → (𝜓𝜓))
3 oplem1.2 . . . . 5 (𝜑 → (𝜃𝜏))
4 ax-1 5 . . . . . 6 (𝜃 → (𝜒𝜃))
5 oplem1.4 . . . . . . 7 (𝜒 → (𝜃𝜏))
65biimprcd 153 . . . . . 6 (𝜏 → (𝜒𝜃))
74, 6jaoi 644 . . . . 5 ((𝜃𝜏) → (𝜒𝜃))
83, 7syl 14 . . . 4 (𝜑 → (𝜒𝜃))
9 oplem1.3 . . . 4 (𝜓𝜃)
108, 9syl6ibr 155 . . 3 (𝜑 → (𝜒𝜓))
112, 10jaod 645 . 2 (𝜑 → ((𝜓𝜒) → 𝜓))
121, 11mpd 13 1 (𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wo 637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  preqr1g  3562  preqr1  3564
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