Theorem elinintab 37881

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

Assertion

Ref	Expression
elinintab	|- ( A e. ( B i^i |^| { x | ph } ) <-> ( A e. B /\ A. x ( ph -> A e. x ) ) )

Distinct variable group:   x, A

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

Proof of Theorem elinintab

Step	Hyp	Ref	Expression
1		elin 3796	. 2 |- ( A e. ( B i^i |^| { x | ph } ) <-> ( A e. B /\ A e. |^| { x | ph } ) )
2		elintabg 37880	. . 3 |- ( A e. B -> ( A e. |^| { x | ph } <-> A. x ( ph -> A e. x ) ) )
3	2	pm5.32i 669	. 2 |- ( ( A e. B /\ A e. |^| { x | ph } ) <-> ( A e. B /\ A. x ( ph -> A e. x ) ) )
4	1, 3	bitri 264	1 |- ( A e. ( B i^i |^| { x | ph } ) <-> ( A e. B /\ A. x ( ph -> A e. x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   i^i cin 3573   |^|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-in 3581  df-int 4476

This theorem is referenced by:  inintabss  37884  inintabd  37885  elcnvcnvintab  37888