Theorem ineq1 3451

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 2414	. . . 4 |- (A = B -> (x e. A <-> x e. B))
2	1	anbi1d 685	. . 3 |- (A = B -> ((x e. A /\ x e. C) <-> (x e. B /\ x e. C)))
3		elin 3220	. . 3 |- (x e. (A i^i C) <-> (x e. A /\ x e. C))
4		elin 3220	. . 3 |- (x e. (B i^i C) <-> (x e. B /\ x e. C))
5	2, 3, 4	3bitr4g 279	. 2 |- (A = B -> (x e. (A i^i C) <-> x e. (B i^i C)))
6	5	eqrdv 2351	1 |- (A = B -> (A i^i C) = (B i^i C))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710   i^i cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214

This theorem is referenced by:  ineq2  3452  ineq12  3453  ineq1i  3454  ineq1d  3457  unineq  3506  dfrab3ss  3534  intprg  3961  eladdci  4400  addcass  4416  nndisjeq  4430  reseq1  4929  brdisjg  5822  qsdisj  5996