Theorem ineq1 3234

Description: Equality theorem for intersection of two classes. (Contributed by NM, 14-Dec-1993.)

Assertion

Ref	Expression
ineq1	|- ( A = B -> ( A i^i C ) = ( B i^i C ) )

Proof of Theorem ineq1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2176	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	anbi1d 458	. . 3 |- ( A = B -> ( ( x e. A /\ x e. C ) <-> ( x e. B /\ x e. C ) ) )
3		elin 3223	. . 3 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3223	. . 3 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
5	2, 3, 4	3bitr4g 222	. 2 |- ( A = B -> ( x e. ( A i^i C ) <-> x e. ( B i^i C ) ) )
6	5	eqrdv 2111	1 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1312   e. wcel 1461   i^i cin 3034

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-in 3041

This theorem is referenced by:  ineq2  3235  ineq12  3236  ineq1i  3237  ineq1d  3240  dfrab3ss  3318  intprg  3768  inex1g  4022  reseq1  4769  fiintim  6768  uzin2  10645  elrestr  11965  inopn  12007  isbasisg  12048  basis1  12051  basis2  12052  tgval  12055  ntrfval  12106  tgrest  12175  restco  12180  restsn  12186  restopnb  12187  txrest  12281  metrest  12489  qtopbasss  12504  bdinex1g  12782