Theorem csbie2 3191

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)

Hypotheses

Ref	Expression
csbie2t.1	|- A e. _V
csbie2t.2	|- B e. _V
csbie2.3	|- ( ( x = A /\ y = B ) -> C = D )

Assertion

Ref	Expression
csbie2	|- [_ A / x ]_ [_ B / y ]_ C = D

Distinct variable groups:   x, y, A   x, B, y   x, D, y

Allowed substitution hints:   C( x, y)

Proof of Theorem csbie2

Step	Hyp	Ref	Expression
1		csbie2.3	. . 3 |- ( ( x = A /\ y = B ) -> C = D )
2	1	gen2 1499	. 2 |- A. x A. y ( ( x = A /\ y = B ) -> C = D )
3		csbie2t.1	. . 3 |- A e. _V
4		csbie2t.2	. . 3 |- B e. _V
5	3, 4	csbie2t 3190	. 2 |- ( A. x A. y ( ( x = A /\ y = B ) -> C = D ) -> [_ A / x ]_ [_ B / y ]_ C = D )
6	2, 5	ax-mp 5	1 |- [_ A / x ]_ [_ B / y ]_ C = D

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1396   = wceq 1398   e. wcel 2205   _Vcvv 2815   [_csb 3141

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-sbc 3046  df-csb 3142

This theorem is referenced by:  fsumcnv  12148  fprodcnv  12336  dfrhm2  14399  vtxdgfval  16409