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Theorem weeq12d 42495
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 5670 . . 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 5671 . . 3 (𝐴 = 𝐵 → (𝑆 We 𝐴𝑆 We 𝐵))
64, 5syl 17 . 2 (𝜑 → (𝑆 We 𝐴𝑆 We 𝐵))
73, 6bitrd 278 1 (𝜑 → (𝑅 We 𝐴𝑆 We 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533   We wwe 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-v 3475  df-in 3956  df-ss 3966  df-br 5153  df-po 5594  df-so 5595  df-fr 5637  df-we 5639
This theorem is referenced by:  fnwe2lem1  42505  aomclem1  42509  aomclem4  42512  aomclem5  42513  aomclem6  42514
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