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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  6303  axcontlem4  25647  cusgrasize  26006  clwwlkf1  26324  eigorthi  28080  expdiophlem2  36607  pwssplit4  36677  fmtnoodd  39983  usgredg4  40444  cusgrsize  40670  uspgr2wlkeqi  40856  clwwlksf1  41224
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