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Theorem weeq12d 38562
 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 5343 . . 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 5344 . . 3 (𝐴 = 𝐵 → (𝑆 We 𝐴𝑆 We 𝐵))
64, 5syl 17 . 2 (𝜑 → (𝑆 We 𝐴𝑆 We 𝐵))
73, 6bitrd 271 1 (𝜑 → (𝑅 We 𝐴𝑆 We 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   = wceq 1601   We wwe 5313 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-ral 3094  df-rex 3095  df-in 3798  df-ss 3805  df-br 4887  df-po 5274  df-so 5275  df-fr 5314  df-we 5316 This theorem is referenced by:  fnwe2lem1  38572  aomclem1  38576  aomclem4  38579  aomclem5  38580  aomclem6  38581
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