Theorem opelopabsbALT 4014

Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4015, 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 1594	. . 3 |- ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( <. z , w >. = <. x , y >. /\ ph ) )
2		vex 2604	. . . . . . 7 |- z e. _V
3		vex 2604	. . . . . . 7 |- w e. _V
4	2, 3	opth 3992	. . . . . 6 |- ( <. z , w >. = <. x , y >. <-> ( z = x /\ w = y ) )
5		equcom 1633	. . . . . . 7 |- ( z = x <-> x = z )
6		equcom 1633	. . . . . . 7 |- ( w = y <-> y = w )
7	5, 6	anbi12ci 448	. . . . . 6 |- ( ( z = x /\ w = y ) <-> ( y = w /\ x = z ) )
8	4, 7	bitri 182	. . . . 5 |- ( <. z , w >. = <. x , y >. <-> ( y = w /\ x = z ) )
9	8	anbi1i 445	. . . 4 |- ( ( <. z , w >. = <. x , y >. /\ ph ) <-> ( ( y = w /\ x = z ) /\ ph ) )
10	9	2exbii 1537	. . 3 |- ( E. y E. x ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
11	1, 10	bitri 182	. 2 |- ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
12		elopab 4013	. 2 |- ( <. z , w >. e. { <. x , y >. | ph } <-> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) )
13		2sb5 1900	. 2 |- ( [ w / y ] [ z / x ] ph <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
14	11, 12, 13	3bitr4i 210	1 |- ( <. z , w >. e. { <. x , y >. | ph } <-> [ w / y ] [ z / x ] ph )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   [wsb 1685   <.cop 3401   {copab 3838

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840

This theorem is referenced by:  inopab  4486  cnvopab  4746