Theorem elintd 39245

Description: Membership in class intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

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

Assertion

Ref	Expression
elintd	|- ( ph -> A e. |^| B )

Distinct variable groups:   x, A   x, B

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

Proof of Theorem elintd

Step	Hyp	Ref	Expression
1		elintd.1	. . 3 |- F/ x ph
2		elintd.3	. . . 4 |- ( ( ph /\ x e. B ) -> A e. x )
3	2	ex 450	. . 3 |- ( ph -> ( x e. B -> A e. x ) )
4	1, 3	ralrimi 2957	. 2 |- ( ph -> A. x e. B A e. x )
5		elintd.2	. . 3 |- ( ph -> A e. V )
6		elintg 4483	. . 3 |- ( A e. V -> ( A e. |^| B <-> A. x e. B A e. x ) )
7	5, 6	syl 17	. 2 |- ( ph -> ( A e. |^| B <-> A. x e. B A e. x ) )
8	4, 7	mpbird 247	1 |- ( ph -> A e. |^| B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   F/wnf 1708   e. wcel 1990   A.wral 2912   |^|cint 4475

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-int 4476

This theorem is referenced by:  ssuniint  39250  elintdv  39251