Theorem opelopabsbALT 3996

Description: The law of concretion in terms of substitutions. Less general than opelopabsb 3997, 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 1554	. . 3 |- ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( <. z , w >. = <. x , y >. /\ ph ) )
2		vex 2560	. . . . . . 7 |- z e. _V
3		vex 2560	. . . . . . 7 |- w e. _V
4	2, 3	opth 3974	. . . . . 6 |- ( <. z , w >. = <. x , y >. <-> ( z = x /\ w = y ) )
5		equcom 1593	. . . . . . 7 |- ( z = x <-> x = z )
6		equcom 1593	. . . . . . 7 |- ( w = y <-> y = w )
7	5, 6	anbi12ci 434	. . . . . 6 |- ( ( z = x /\ w = y ) <-> ( y = w /\ x = z ) )
8	4, 7	bitri 173	. . . . 5 |- ( <. z , w >. = <. x , y >. <-> ( y = w /\ x = z ) )
9	8	anbi1i 431	. . . 4 |- ( ( <. z , w >. = <. x , y >. /\ ph ) <-> ( ( y = w /\ x = z ) /\ ph ) )
10	9	2exbii 1497	. . 3 |- ( E. y E. x ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
11	1, 10	bitri 173	. 2 |- ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
12		elopab 3995	. 2 |- ( <. z , w >. e. { <. x , y >. | ph } <-> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) )
13		2sb5 1859	. 2 |- ( [ w / y ] [ z / x ] ph <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
14	11, 12, 13	3bitr4i 201	1 |- ( <. z , w >. e. { <. x , y >. | ph } <-> [ w / y ] [ z / x ] ph )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   [wsb 1645   <.cop 3378   {copab 3817

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819

This theorem is referenced by:  inopab  4468  cnvopab  4726