Theorem iineq2d 3961

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 2579	. 2 |- ( ph -> A. x e. A B = C )
5		iineq2 3958	. 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 1373   F/wnf 1484   e. wcel 2178   A.wral 2486   |^|_ciin 3942

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-ral 2491  df-iin 3944

This theorem is referenced by:  iineq2dv  3963