Theorem eliind 39240

Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
eliind.a	|- ( ph -> A e. |^|_ x e. B C )
eliind.k	|- ( ph -> K e. B )
eliind.d	|- ( x = K -> ( A e. C <-> A e. D ) )

Assertion

Ref	Expression
eliind	|- ( ph -> A e. D )

Distinct variable groups:   x, A   x, B   x, D   x, K

Allowed substitution hints:   ph( x)   C( x)

Proof of Theorem eliind

Step	Hyp	Ref	Expression
1		eliind.k	. 2 |- ( ph -> K e. B )
2		eliind.a	. . 3 |- ( ph -> A e. |^|_ x e. B C )
3		eliin 4525	. . . 4 |- ( A e. |^|_ x e. B C -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )
4	2, 3	syl 17	. . 3 |- ( ph -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )
5	2, 4	mpbid 222	. 2 |- ( ph -> A. x e. B A e. C )
6		eliind.d	. . 3 |- ( x = K -> ( A e. C <-> A e. D ) )
7	6	rspcva 3307	. 2 |- ( ( K e. B /\ A. x e. B A e. C ) -> A e. D )
8	1, 5, 7	syl2anc 693	1 |- ( ph -> A e. D )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-iin 4523

This theorem is referenced by:  iooiinioc  39783  hspdifhsp  40830  smflimlem3  40981  smfsuplem1  41017  smflimsuplem4  41029