Theorem elintabg 37880

Description: Two ways of saying a set is an element of the intersection of a class. (Contributed by RP, 13-Aug-2020.)

Assertion

Ref	Expression
elintabg	|- ( A e. V -> ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) )

Distinct variable group:   x, A

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

Proof of Theorem elintabg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elintg 4483	. 2 |- ( A e. V -> ( A e. |^| { x | ph } <-> A. y e. { x | ph } A e. y ) )
2		eleq2 2690	. . 3 |- ( y = x -> ( A e. y <-> A e. x ) )
3	2	ralab2 3371	. 2 |- ( A. y e. { x | ph } A e. y <-> A. x ( ph -> A e. x ) )
4	1, 3	syl6bb 276	1 |- ( A e. V -> ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   {cab 2608   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:  elinintab  37881