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Theorem ineq1 3108
Description: Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)
Assertion
Ref Expression
ineq1 (A = B → (A𝐶) = (B𝐶))

Proof of Theorem ineq1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2083 . . . 4 (A = B → (x Ax B))
21anbi1d 441 . . 3 (A = B → ((x A x 𝐶) ↔ (x B x 𝐶)))
3 elin 3103 . . 3 (x (A𝐶) ↔ (x A x 𝐶))
4 elin 3103 . . 3 (x (B𝐶) ↔ (x B x 𝐶))
52, 3, 43bitr4g 212 . 2 (A = B → (x (A𝐶) ↔ x (B𝐶)))
65eqrdv 2020 1 (A = B → (A𝐶) = (B𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374  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:  ineq2  3109  ineq12  3110  ineq1i  3111  ineq1d  3114  dfrab3ss  3192  intprg  3622  inex1g  3867  reseq1  4533  bdinex1g  7124
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