Theorem eloprabg 5728

Description: The law of concretion for operation class abstraction. Compare elopab 4083. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
eloprabg.1	|- ( x = A -> ( ph <-> ps ) )
eloprabg.2	|- ( y = B -> ( ps <-> ch ) )
eloprabg.3	|- ( z = C -> ( ch <-> th ) )

Assertion

Ref	Expression
eloprabg	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> th ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   th, x, y, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z)   ch( x, y, z)   V( x, y, z)   W( x, y, z)   X( x, y, z)

Proof of Theorem eloprabg

Step	Hyp	Ref	Expression
1		eloprabg.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
2		eloprabg.2	. . 3 |- ( y = B -> ( ps <-> ch ) )
3		eloprabg.3	. . 3 |- ( z = C -> ( ch <-> th ) )
4	1, 2, 3	syl3an9b 1246	. 2 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> th ) )
5	4	eloprabga 5727	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> th ) )

Syntax hints:   -> wi 4   <-> wb 103   /\ w3a 924   = wceq 1289   e. wcel 1438   <.cop 3447   {coprab 5645

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-oprab 5648

This theorem is referenced by:  ov  5756  ovg  5775