Theorem iineq2d 3906

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 2548	. 2 |- ( ph -> A. x e. A B = C )
5		iineq2 3903	. 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 1353   F/wnf 1460   e. wcel 2148   A.wral 2455   |^|_ciin 3887

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-iin 3889

This theorem is referenced by:  iineq2dv  3908