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Theorem eqeq12i 2069
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 2068 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
41, 2, 3mp2an 410 1 (𝐴 = 𝐶𝐵 = 𝐷)
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-cleq 2049
This theorem is referenced by:  rabbi  2504  sbceqg  2894  preqr2g  3566  preqr2  3568  otth  4007  rncoeq  4633  eqfnov  5635  mpt22eqb  5638  f1o2ndf1  5877  ecopovsym  6233  sq11i  9509
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