Theorem csbie2 3563

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 1723	. 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 3562	. 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 384   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  fsumcnv  14504  fprodcnv  14713  dfrhm2  18717  mamufval  20191  mvmulfval  20348  vtxdgfval  26363  rnghmval  41891