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Theorem weeq12d 39837
Description: Equality deduction for well-orders. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
weeq12d.l (𝜑𝑅 = 𝑆)
weeq12d.r (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
weeq12d (𝜑 → (𝑅 We 𝐴𝑆 We 𝐵))

Proof of Theorem weeq12d
StepHypRef Expression
1 weeq12d.l . . 3 (𝜑𝑅 = 𝑆)
2 weeq1 5530 . . 3 (𝑅 = 𝑆 → (𝑅 We 𝐴𝑆 We 𝐴))
31, 2syl 17 . 2 (𝜑 → (𝑅 We 𝐴𝑆 We 𝐴))
4 weeq12d.r . . 3 (𝜑𝐴 = 𝐵)
5 weeq2 5531 . . 3 (𝐴 = 𝐵 → (𝑆 We 𝐴𝑆 We 𝐵))
64, 5syl 17 . 2 (𝜑 → (𝑆 We 𝐴𝑆 We 𝐵))
73, 6bitrd 282 1 (𝜑 → (𝑅 We 𝐴𝑆 We 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538   We wwe 5500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3138  df-rex 3139  df-v 3482  df-in 3926  df-ss 3936  df-br 5053  df-po 5461  df-so 5462  df-fr 5501  df-we 5503
This theorem is referenced by:  fnwe2lem1  39847  aomclem1  39851  aomclem4  39854  aomclem5  39855  aomclem6  39856
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