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Theorem eqeqan12rd 2628
Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)
Hypotheses
Ref Expression
eqeqan12rd.1 (𝜑𝐴 = 𝐵)
eqeqan12rd.2 (𝜓𝐶 = 𝐷)
Assertion
Ref Expression
eqeqan12rd ((𝜓𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))

Proof of Theorem eqeqan12rd
StepHypRef Expression
1 eqeqan12rd.1 . . 3 (𝜑𝐴 = 𝐵)
2 eqeqan12rd.2 . . 3 (𝜓𝐶 = 𝐷)
31, 2eqeqan12d 2626 . 2 ((𝜑𝜓) → (𝐴 = 𝐶𝐵 = 𝐷))
43ancoms 468 1 ((𝜓𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475
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-ext 2590
This theorem depends on definitions:  df-bi 196  df-an 385  df-cleq 2603
This theorem is referenced by:  fmptco  6288  axcontlem4  25593  cusgrasize  25800  clwwlkf1  26118  eigorthi  27874  expdiophlem2  36401  pwssplit4  36471  fmtnoodd  39778  usgredg4  40436  cusgrsize  40662  uspgr2wlkeqi  40848  clwwlksf1  41216
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