Theorem opelopabsbALT 4293

Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4294, 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 1678	. . 3 |- ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( <. z , w >. = <. x , y >. /\ ph ) )
2		vex 2766	. . . . . . 7 |- z e. _V
3		vex 2766	. . . . . . 7 |- w e. _V
4	2, 3	opth 4270	. . . . . 6 |- ( <. z , w >. = <. x , y >. <-> ( z = x /\ w = y ) )
5		equcom 1720	. . . . . . 7 |- ( z = x <-> x = z )
6		equcom 1720	. . . . . . 7 |- ( w = y <-> y = w )
7	5, 6	anbi12ci 461	. . . . . 6 |- ( ( z = x /\ w = y ) <-> ( y = w /\ x = z ) )
8	4, 7	bitri 184	. . . . 5 |- ( <. z , w >. = <. x , y >. <-> ( y = w /\ x = z ) )
9	8	anbi1i 458	. . . 4 |- ( ( <. z , w >. = <. x , y >. /\ ph ) <-> ( ( y = w /\ x = z ) /\ ph ) )
10	9	2exbii 1620	. . 3 |- ( E. y E. x ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
11	1, 10	bitri 184	. 2 |- ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
12		elopab 4292	. 2 |- ( <. z , w >. e. { <. x , y >. | ph } <-> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) )
13		2sb5 2002	. 2 |- ( [ w / y ] [ z / x ] ph <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
14	11, 12, 13	3bitr4i 212	1 |- ( <. z , w >. e. { <. x , y >. | ph } <-> [ w / y ] [ z / x ] ph )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1506   [wsb 1776   e. wcel 2167   <.cop 3625   {copab 4093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095

This theorem is referenced by:  inopab  4798  cnvopab  5071