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Theorem ineq1d 3114
Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
Hypothesis
Ref Expression
ineq1d.1 (φA = B)
Assertion
Ref Expression
ineq1d (φ → (A𝐶) = (B𝐶))

Proof of Theorem ineq1d
StepHypRef Expression
1 ineq1d.1 . 2 (φA = B)
2 ineq1 3108 . 2 (A = B → (A𝐶) = (B𝐶))
31, 2syl 14 1 (φ → (A𝐶) = (B𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228  cin 2893
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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901
This theorem is referenced by:  diftpsn3  3479  ordpwsucexmid  4230  riinint  4520  fnresdisj  4935  fnimadisj  4945  ecinxp  6092
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