Theorem csbco3g 2961

Description: Composition of two class substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

Hypothesis

Ref	Expression
sbcco3g.1	|- ( x = A -> B = C )

Assertion

Ref	Expression
csbco3g	|- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ D )

Distinct variable groups:   x, A   x, C   x, D

Allowed substitution hints:   A( y)   B( x, y)   C( y)   D( y)   V( x, y)

Proof of Theorem csbco3g

Step	Hyp	Ref	Expression
1		csbnestg 2957	. 2 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ [_ A / x ]_ B / y ]_ D )
2		elex 2611	. . . 4 |- ( A e. V -> A e. _V )
3		nfcvd 2221	. . . . 5 |- ( A e. _V -> F/_ x C )
4		sbcco3g.1	. . . . 5 |- ( x = A -> B = C )
5	3, 4	csbiegf 2947	. . . 4 |- ( A e. _V -> [_ A / x ]_ B = C )
6	2, 5	syl 14	. . 3 |- ( A e. V -> [_ A / x ]_ B = C )
7	6	csbeq1d 2915	. 2 |- ( A e. V -> [_ [_ A / x ]_ B / y ]_ D = [_ C / y ]_ D )
8	1, 7	eqtrd 2114	1 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ D )

Syntax hints:   -> wi 4   = wceq 1285   e. wcel 1434   _Vcvv 2602   [_csb 2909

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910

This theorem is referenced by: (None)