Theorem eloprabg 6010

Description: The law of concretion for operation class abstraction. Compare elopab 4292. (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 1321	. 2 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> th ) )
5	4	eloprabga 6009	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> th ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   <.cop 3625   {coprab 5923

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-oprab 5926

This theorem is referenced by:  ov  6042  ovg  6062