Theorem opelopabsbALT 4244

Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4245, 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 1657	. . 3 |- ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( <. z , w >. = <. x , y >. /\ ph ) )
2		vex 2733	. . . . . . 7 |- z e. _V
3		vex 2733	. . . . . . 7 |- w e. _V
4	2, 3	opth 4222	. . . . . 6 |- ( <. z , w >. = <. x , y >. <-> ( z = x /\ w = y ) )
5		equcom 1699	. . . . . . 7 |- ( z = x <-> x = z )
6		equcom 1699	. . . . . . 7 |- ( w = y <-> y = w )
7	5, 6	anbi12ci 458	. . . . . 6 |- ( ( z = x /\ w = y ) <-> ( y = w /\ x = z ) )
8	4, 7	bitri 183	. . . . 5 |- ( <. z , w >. = <. x , y >. <-> ( y = w /\ x = z ) )
9	8	anbi1i 455	. . . 4 |- ( ( <. z , w >. = <. x , y >. /\ ph ) <-> ( ( y = w /\ x = z ) /\ ph ) )
10	9	2exbii 1599	. . 3 |- ( E. y E. x ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
11	1, 10	bitri 183	. 2 |- ( E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
12		elopab 4243	. 2 |- ( <. z , w >. e. { <. x , y >. | ph } <-> E. x E. y ( <. z , w >. = <. x , y >. /\ ph ) )
13		2sb5 1976	. 2 |- ( [ w / y ] [ z / x ] ph <-> E. y E. x ( ( y = w /\ x = z ) /\ ph ) )
14	11, 12, 13	3bitr4i 211	1 |- ( <. z , w >. e. { <. x , y >. | ph } <-> [ w / y ] [ z / x ] ph )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   [wsb 1755   e. wcel 2141   <.cop 3586   {copab 4049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051

This theorem is referenced by:  inopab  4743  cnvopab  5012