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Theorem oplem1 1006
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 → (𝜓𝜒))
oplem1.2 (𝜑 → (𝜃𝜏))
oplem1.3 (𝜓𝜃)
oplem1.4 (𝜒 → (𝜃𝜏))
Assertion
Ref Expression
oplem1 (𝜑𝜓)

Proof of Theorem oplem1
StepHypRef Expression
1 oplem1.3 . . . . . . 7 (𝜓𝜃)
21notbii 310 . . . . . 6 𝜓 ↔ ¬ 𝜃)
3 oplem1.1 . . . . . . 7 (𝜑 → (𝜓𝜒))
43ord 392 . . . . . 6 (𝜑 → (¬ 𝜓𝜒))
52, 4syl5bir 233 . . . . 5 (𝜑 → (¬ 𝜃𝜒))
6 oplem1.2 . . . . . 6 (𝜑 → (𝜃𝜏))
76ord 392 . . . . 5 (𝜑 → (¬ 𝜃𝜏))
85, 7jcad 555 . . . 4 (𝜑 → (¬ 𝜃 → (𝜒𝜏)))
9 oplem1.4 . . . . 5 (𝜒 → (𝜃𝜏))
109biimpar 502 . . . 4 ((𝜒𝜏) → 𝜃)
118, 10syl6 35 . . 3 (𝜑 → (¬ 𝜃𝜃))
1211pm2.18d 124 . 2 (𝜑𝜃)
1312, 1sylibr 224 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384
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 197  df-or 385  df-an 386
This theorem is referenced by:  preq1b  4368  preqr1OLD  4371
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