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Theorem eqeq12i 2053
 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 2052 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))
41, 2, 3mp2an 402 1 (𝐴 = 𝐶𝐵 = 𝐷)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1243 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-cleq 2033 This theorem is referenced by:  rabbi  2487  sbceqg  2866  preqr2g  3538  preqr2  3540  otth  3979  rncoeq  4605  eqfnov  5607  mpt22eqb  5610  f1o2ndf1  5849  ecopovsym  6202  sq11i  9343
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