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Theorem ineq1 3100
 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 2075 . . . 4 (A = B → (x Ax B))
21anbi1d 438 . . 3 (A = B → ((x A x 𝐶) ↔ (x B x 𝐶)))
3 elin 3095 . . 3 (x (A𝐶) ↔ (x A x 𝐶))
4 elin 3095 . . 3 (x (B𝐶) ↔ (x B x 𝐶))
52, 3, 43bitr4g 212 . 2 (A = B → (x (A𝐶) ↔ x (B𝐶)))
65eqrdv 2012 1 (A = B → (A𝐶) = (B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1224   ∈ wcel 1367   ∩ cin 2885 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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996 This theorem depends on definitions:  df-bi 110  df-tru 1227  df-nf 1324  df-sb 1620  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-v 2529  df-in 2893 This theorem is referenced by:  ineq2  3101  ineq12  3102  ineq1i  3103  ineq1d  3106  dfrab3ss  3184  intprg  3612  inex1g  3857  reseq1  4522  bdinex1g  8354
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