Theorem eloprabg 6035

Description: The law of concretion for operation class abstraction. Compare elopab 4305. (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 1323	. 2 |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> th ) )
5	4	eloprabga 6034	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 981   = wceq 1373   e. wcel 2176   <.cop 3636   {coprab 5947

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-oprab 5950

This theorem is referenced by:  ov  6067  ovg  6087