Theorem ineq1 3265

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 2201	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	anbi1d 460	. . 3 |- ( A = B -> ( ( x e. A /\ x e. C ) <-> ( x e. B /\ x e. C ) ) )
3		elin 3254	. . 3 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3254	. . 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 2135	1 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   i^i cin 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072

This theorem is referenced by:  ineq2  3266  ineq12  3267  ineq1i  3268  ineq1d  3271  dfrab3ss  3349  intprg  3799  inex1g  4059  reseq1  4808  fiintim  6810  uzin2  10752  elrestr  12117  inopn  12159  isbasisg  12200  basis1  12203  basis2  12204  tgval  12207  ntrfval  12258  tgrest  12327  restco  12332  restsn  12338  restopnb  12339  txrest  12434  metrest  12664  qtopbasss  12679  bdinex1g  13088