Theorem bropabid 34128

Description: Lemma for ~? inxptxp . (Contributed by Peter Mazsa, 24-Nov-2018.)

Hypothesis

Ref	Expression
bropabid.1	|- R = { <. x , y >. | ph }

Assertion

Ref	Expression
bropabid	|- ( x R y <-> ph )

Proof of Theorem bropabid

Step	Hyp	Ref	Expression
1		bropabid.1	. . 3 |- R = { <. x , y >. | ph }
2	1	breqi 4659	. 2 |- ( x R y <-> x { <. x , y >. | ph } y )
3		df-br 4654	. 2 |- ( x { <. x , y >. | ph } y <-> <. x , y >. e. { <. x , y >. | ph } )
4		opabid 4982	. 2 |- ( <. x , y >. e. { <. x , y >. | ph } <-> ph )
5	2, 3, 4	3bitri 286	1 |- ( x R y <-> ph )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   {copab 4712

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  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-br 4654  df-opab 4713

This theorem is referenced by: (None)