Theorem eliind2 39313

Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
eliind2.1	|- F/ x ph
eliind2.2	|- ( ph -> A e. V )
eliind2.3	|- ( ( ph /\ x e. B ) -> A e. C )

Assertion

Ref	Expression
eliind2	|- ( ph -> A e. |^|_ x e. B C )

Distinct variable group:   x, A

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

Proof of Theorem eliind2

Step	Hyp	Ref	Expression
1		eliind2.1	. . 3 |- F/ x ph
2		eliind2.3	. . . 4 |- ( ( ph /\ x e. B ) -> A e. C )
3	2	ex 450	. . 3 |- ( ph -> ( x e. B -> A e. C ) )
4	1, 3	ralrimi 2957	. 2 |- ( ph -> A. x e. B A e. C )
5		eliind2.2	. . 3 |- ( ph -> A e. V )
6		eliin 4525	. . 3 |- ( A e. V -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )
7	5, 6	syl 17	. 2 |- ( ph -> ( A e. |^|_ x e. B C <-> A. x e. B A e. C ) )
8	4, 7	mpbird 247	1 |- ( ph -> A e. |^|_ x e. B C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   F/wnf 1708   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: (None)