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Theorem eqeqan12rd 2749
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 2748 . 2 ((𝜑𝜓) → (𝐴 = 𝐶𝐵 = 𝐷))
43ancoms 458 1 ((𝜓𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-cleq 2726
This theorem is referenced by:  fmptco  7148  axcontlem4  28996  usgredg4  29248  cusgrsize  29486  uspgr2wlkeqi  29680  clwwlkf1  30077  eigorthi  31865  goeleq12bg  35333  expdiophlem2  43010  pwssplit4  43077  fcoresf1  47018  prproropf1olem4  47430  fmtnoodd  47457  isgrim  47805
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