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Theorem eqeqan12rd 2753
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 2752 . 2 ((𝜑𝜓) → (𝐴 = 𝐶𝐵 = 𝐷))
43ancoms 462 1 ((𝜓𝜑) → (𝐴 = 𝐶𝐵 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-cleq 2730
This theorem is referenced by:  fmptco  6962  axcontlem4  27082  usgredg4  27329  cusgrsize  27566  uspgr2wlkeqi  27759  clwwlkf1  28156  eigorthi  29942  goeleq12bg  33047  expdiophlem2  40575  pwssplit4  40645  fcoresf1  44263  prproropf1olem4  44659  fmtnoodd  44686
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