Theorem brintclab 13742

Description: Two ways to express a binary relation which is the intersection of a class. (Contributed by RP, 4-Apr-2020.)

Assertion

Ref	Expression
brintclab	|- ( A |^| { x | ph } B <-> A. x ( ph -> <. A , B >. e. x ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem brintclab

Step	Hyp	Ref	Expression
1		df-br 4654	. 2 |- ( A |^| { x | ph } B <-> <. A , B >. e. |^| { x | ph } )
2		opex 4932	. . 3 |- <. A , B >. e. _V
3	2	elintab 4487	. 2 |- ( <. A , B >. e. |^| { x | ph } <-> A. x ( ph -> <. A , B >. e. x ) )
4	1, 3	bitri 264	1 |- ( A |^| { x | ph } B <-> A. x ( ph -> <. A , B >. e. x ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   {cab 2608   <.cop 4183   |^|cint 4475   class class class wbr 4653

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-int 4476  df-br 4654

This theorem is referenced by:  brtrclfv  13743