Theorem iineq2d 3698

Description: Equality deduction for indexed intersection. (Contributed by NM, 7-Dec-2011.)

Hypotheses

Ref	Expression
iineq2d.1	|- F/ x ph
iineq2d.2	|- ( ( ph /\ x e. A ) -> B = C )

Assertion

Ref	Expression
iineq2d	|- ( ph -> |^|_ x e. A B = |^|_ x e. A C )

Proof of Theorem iineq2d

Step	Hyp	Ref	Expression
1		iineq2d.1	. . 3 |- F/ x ph
2		iineq2d.2	. . . 4 |- ( ( ph /\ x e. A ) -> B = C )
3	2	ex 113	. . 3 |- ( ph -> ( x e. A -> B = C ) )
4	1, 3	ralrimi 2432	. 2 |- ( ph -> A. x e. A B = C )
5		iineq2 3695	. 2 |- ( A. x e. A B = C -> |^|_ x e. A B = |^|_ x e. A C )
6	4, 5	syl 14	1 |- ( ph -> |^|_ x e. A B = |^|_ x e. A C )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   F/wnf 1389   e. wcel 1433   A.wral 2348   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-iin 3681

This theorem is referenced by:  iineq2dv  3700