Theorem eloprabg 5966

Description: The law of concretion for operation class abstraction. Compare elopab 4260. (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 1310	. 2 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> th ) )
5	4	eloprabga 5965	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 978   = wceq 1353   e. wcel 2148   <.cop 3597   {coprab 5879

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-oprab 5882

This theorem is referenced by:  ov  5997  ovg  6016