Theorem iineq2d 3918

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 115	. . 3 |- ( ph -> ( x e. A -> B = C ) )
4	1, 3	ralrimi 2558	. 2 |- ( ph -> A. x e. A B = C )
5		iineq2 3915	. 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 104   = wceq 1363   F/wnf 1470   e. wcel 2158   A.wral 2465   |^|_ciin 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-ral 2470  df-iin 3901

This theorem is referenced by:  iineq2dv  3920