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Theorem freq12d 39970
 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 5493 . . 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 5494 . . 3 (𝐴 = 𝐵 → (𝑆 Fr 𝐴𝑆 Fr 𝐵))
64, 5syl 17 . 2 (𝜑 → (𝑆 Fr 𝐴𝑆 Fr 𝐵))
73, 6bitrd 282 1 (𝜑 → (𝑅 Fr 𝐴𝑆 Fr 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   Fr wfr 5479 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115  df-v 3446  df-in 3891  df-ss 3901  df-br 5034  df-fr 5482 This theorem is referenced by: (None)
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