Theorem ineq1 3311

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 2228	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	anbi1d 461	. . 3 |- ( A = B -> ( ( x e. A /\ x e. C ) <-> ( x e. B /\ x e. C ) ) )
3		elin 3300	. . 3 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3300	. . 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 2162	1 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1342   e. wcel 2135   i^i cin 3110

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-in 3117

This theorem is referenced by:  ineq2  3312  ineq12  3313  ineq1i  3314  ineq1d  3317  dfrab3ss  3395  intprg  3851  inex1g  4112  reseq1  4872  fiintim  6885  uzin2  10915  elrestr  12500  inopn  12542  isbasisg  12583  basis1  12586  basis2  12587  tgval  12590  ntrfval  12641  tgrest  12710  restco  12715  restsn  12721  restopnb  12722  txrest  12817  metrest  13047  qtopbasss  13062  bdinex1g  13618