Theorem ineq1 1638

Description: Equality theorem for intersection of two classes.

Assertion

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

Proof of Theorem ineq1

Step	Hyp	Ref	Expression
1		eleq2 1150	. . . 4 |- (A = B -> (x e. A <-> x e. B))
2	1	anbi1d 469	. . 3 |- (A = B -> ((x e. A /\ x e. C) <-> (x e. B /\ x e. C)))
3		elin 1635	. . 3 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
4		elin 1635	. . 3 |- (x e. (B i^i C) <-> (x e. B /\ x e. C))
5	2, 3, 4	3bitr4g 428	. 2 |- (A = B -> (x e. (A i^i C) <-> x e. (B i^i C)))
6	5	cleqrd 1100	1 |- (A = B -> (A i^i C) = (B i^i C))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092   i^i cin 1486

This theorem is referenced by:  ineq2 1639  ineq12 1640  ineq1i 1641  ineq1d 1644  unineq 1680  inex1g 1699  frc 2172  reseq1 2575  isofrlem 2939  fiint 3445  inf3lema 3460  aceq5lem5 3562  kmlem11 3590  kmlem14 3593  omls 5251  shinclt 5352  shmod 5364  chinclt 5416  chdmm1t 5438  cmbrt 5494  mdbr 5726  dmdbr 5731  dmdi 5732  cvexcht 5763

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491

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