Theorem ineq1 3415

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 2296	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	anbi1d 465	. . 3 |- ( A = B -> ( ( x e. A /\ x e. C ) <-> ( x e. B /\ x e. C ) ) )
3		elin 3402	. . 3 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3402	. . 3 |- ( x e. ( B i^i C ) <-> ( x e. B /\ x e. C ) )
5	2, 3, 4	3bitr4g 223	. 2 |- ( A = B -> ( x e. ( A i^i C ) <-> x e. ( B i^i C ) ) )
6	5	eqrdv 2230	1 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   i^i cin 3210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217

This theorem is referenced by:  ineq2  3416  ineq12  3417  ineq1i  3418  ineq1d  3421  dfrab3ss  3499  intprg  3982  inex1g  4246  reseq1  5032  fiintim  7191  uzin2  11672  ressvalsets  13277  elrestr  13460  tgval  13475  inopn  14868  isbasisg  14909  basis1  14912  basis2  14913  ntrfval  14965  tgrest  15034  restco  15039  restsn  15045  restopnb  15046  txrest  15141  metrest  15371  qtopbasss  15386  bdinex1g  16671