Theorem opelopabsbALT 4984

Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4985, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
opelopabsbALT	|- ( <. z , w >. e. { <. x , y >. | ph } <-> [ w / y ] [ z / x ] ph )

Distinct variable groups:   x, y, z   x, w, y

Allowed substitution hints:   ph( x, y, z, w)

Proof of Theorem opelopabsbALT

Step	Hyp	Ref	Expression
1		excom 2042	. . 3 |- ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( <. z , w >. = <. x , y >. /\ ph ) )
2		vex 3203	. . . . . . 7 |- z e. _V
3		vex 3203	. . . . . . 7 |- w e. _V
4	2, 3	opth 4945	. . . . . 6 |- ( <. z , w >. = <. x , y >. <-> ( z = x /\ w = y ) )
5		equcom 1945	. . . . . . 7 |- ( z = x <-> x = z )
6		equcom 1945	. . . . . . 7 |- ( w = y <-> y = w )
7	5, 6	anbi12ci 734	. . . . . 6 |- ( ( z = x /\ w = y ) <-> ( y = w /\ x = z ) )
8	4, 7	bitri 264	. . . . 5 |- ( <. z , w >. = <. x , y >. <-> ( y = w /\ x = z ) )
9	8	anbi1i 731	. . . 4 |- ( ( <. z , w >. = <. x , y >. /\ ph ) <-> ( ( y = w /\ x = z ) /\ ph ) )
10	9	2exbii 1775	. . 3 |- ( E. y E. x ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
11	1, 10	bitri 264	. 2 |- ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
12		elopab 4983	. 2 |- ( <. z , w >. e. { <. x , y >. | ph } <-> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) )
13		2sb5 2443	. 2 |- ( [ w / y ] [ z / x ] ph <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
14	11, 12, 13	3bitr4i 292	1 |- ( <. z , w >. e. { <. x , y >. | ph } <-> [ w / y ] [ z / x ] ph )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   [wsb 1880   e. wcel 1990   <.cop 4183   {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-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-opab 4713

This theorem is referenced by:  inopab  5252  cnvopab  5533  brabsb2  34147