Theorem iineq2d 3886

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 114	. . 3 |- ( ph -> ( x e. A -> B = C ) )
4	1, 3	ralrimi 2537	. 2 |- ( ph -> A. x e. A B = C )
5		iineq2 3883	. 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 103   = wceq 1343   F/wnf 1448   e. wcel 2136   A.wral 2444   |^|_ciin 3867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-iin 3869

This theorem is referenced by:  iineq2dv  3888