Theorem ineq1 3321

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 2234	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	anbi1d 462	. . 3 |- ( A = B -> ( ( x e. A /\ x e. C ) <-> ( x e. B /\ x e. C ) ) )
3		elin 3310	. . 3 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3310	. . 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 2168	1 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   i^i cin 3120

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127

This theorem is referenced by:  ineq2  3322  ineq12  3323  ineq1i  3324  ineq1d  3327  dfrab3ss  3405  intprg  3864  inex1g  4125  reseq1  4885  fiintim  6906  uzin2  10951  elrestr  12587  inopn  12795  isbasisg  12836  basis1  12839  basis2  12840  tgval  12843  ntrfval  12894  tgrest  12963  restco  12968  restsn  12974  restopnb  12975  txrest  13070  metrest  13300  qtopbasss  13315  bdinex1g  13936