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Theorem eqeq12i 2178
Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
eqeq12i.1 𝐴 = 𝐵
eqeq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
eqeq12i (𝐴 = 𝐶𝐵 = 𝐷)

Proof of Theorem eqeq12i
StepHypRef Expression
1 eqeq12i.1 . 2 𝐴 = 𝐵
2 eqeq12i.2 . 2 𝐶 = 𝐷
3 eqeq12 2177 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
41, 2, 3mp2an 423 1 (𝐴 = 𝐶𝐵 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-4 1497  ax-17 1513  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-cleq 2157
This theorem is referenced by:  rabbi  2641  sbceqg  3057  preqr2g  3742  preqr2  3744  otth  4215  rncoeq  4872  eqfnov  5940  mpo2eqb  5943  f1o2ndf1  6188  ecopovsym  6589  sq11i  10535  pwle2  13739
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