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Theorem eleq12d 2090
Description: Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
Hypotheses
Ref Expression
eleq1d.1 (φA = B)
eleq12d.2 (φ𝐶 = 𝐷)
Assertion
Ref Expression
eleq12d (φ → (A 𝐶B 𝐷))

Proof of Theorem eleq12d
StepHypRef Expression
1 eleq12d.2 . . 3 (φ𝐶 = 𝐷)
21eleq2d 2089 . 2 (φ → (A 𝐶A 𝐷))
3 eleq1d.1 . . 3 (φA = B)
43eleq1d 2088 . 2 (φ → (A 𝐷B 𝐷))
52, 4bitrd 177 1 (φ → (A 𝐶B 𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-17 1400  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-cleq 2015  df-clel 2018
This theorem is referenced by:  cbvraldva2  2515  cbvrexdva2  2516  cdeqel  2737  ru  2740  sbcel12g  2842  cbvralcsf  2885  cbvrexcsf  2886  cbvreucsf  2887  cbvrabcsf  2888  elvvuni  4331  elrnmpt1  4512  smoeq  5827  smores  5829  smores2  5831  iordsmo  5834  nnaordi  5992  nnaordr  5994  ltapig  6198  ltmpig  6199
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