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Theorem freq12d 36627
 Description: Equality deduction for founded relations. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
weeq12d.l (𝜑𝑅 = 𝑆)
weeq12d.r (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
freq12d (𝜑 → (𝑅 Fr 𝐴𝑆 Fr 𝐵))

Proof of Theorem freq12d
StepHypRef Expression
1 weeq12d.l . . 3 (𝜑𝑅 = 𝑆)
2 freq1 5008 . . 3 (𝑅 = 𝑆 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))
31, 2syl 17 . 2 (𝜑 → (𝑅 Fr 𝐴𝑆 Fr 𝐴))
4 weeq12d.r . . 3 (𝜑𝐴 = 𝐵)
5 freq2 5009 . . 3 (𝐴 = 𝐵 → (𝑆 Fr 𝐴𝑆 Fr 𝐵))
64, 5syl 17 . 2 (𝜑 → (𝑆 Fr 𝐴𝑆 Fr 𝐵))
73, 6bitrd 267 1 (𝜑 → (𝑅 Fr 𝐴𝑆 Fr 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   Fr wfr 4994 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-rex 2902  df-in 3547  df-ss 3554  df-br 4584  df-fr 4997 This theorem is referenced by: (None)
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