Theorem brabsb 4986

Description: The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)

Hypothesis

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

Assertion

Ref	Expression
brabsb	|- ( A R B <-> [. A / x ]. [. B / y ]. ph )

Distinct variable groups:   x, y   x, B

Allowed substitution hints:   ph( x, y)   A( x, y)   B( y)   R( x, y)

Proof of Theorem brabsb

Step	Hyp	Ref	Expression
1		df-br 4654	. 2 |- ( A R B <-> <. A , B >. e. R )
2		brabsb.1	. . 3 |- R = { <. x , y >. | ph }
3	2	eleq2i 2693	. 2 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ph } )
4		opelopabsb 4985	. 2 |- ( <. A , B >. e. { <. x , y >. | ph } <-> [. A / x ]. [. B / y ]. ph )
5	1, 3, 4	3bitri 286	1 |- ( A R B <-> [. A / x ]. [. B / y ]. ph )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   [.wsbc 3435   <.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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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:  eqerlem  7776