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Theorem eleq12d 2421
 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 2420 . 2 (φ → (A CA D))
3 eleq1d.1 . . 3 (φA = B)
43eleq1d 2419 . 2 (φ → (A DB D))
52, 4bitrd 244 1 (φ → (A CB D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349 This theorem is referenced by:  cbvraldva2  2839  cbvrexdva2  2840  ru  3045  sbcel12g  3151  cbvralcsf  3198  cbvreucsf  3200  cbvrabcsf  3201  nenpw1pwlem2  6085  nmembers1  6271
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