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Theorem eqeq12i 2050
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 2049 . 2 ((A = B 𝐶 = 𝐷) → (A = 𝐶B = 𝐷))
41, 2, 3mp2an 402 1 (A = 𝐶B = 𝐷)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242
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 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-cleq 2030
This theorem is referenced by:  rabbi  2481  sbceqg  2860  preqr2g  3529  preqr2  3531  otth  3970  rncoeq  4548  eqfnov  5549  mpt22eqb  5552  f1o2ndf1  5791  ecopovsym  6138  sq11i  8976
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