Theorem csbco3g 3130

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 3126	. 2 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ [_ A / x ]_ B / y ]_ D )
2		elex 2763	. . . 4 |- ( A e. V -> A e. _V )
3		nfcvd 2333	. . . . 5 |- ( A e. _V -> F/_ x C )
4		sbcco3g.1	. . . . 5 |- ( x = A -> B = C )
5	3, 4	csbiegf 3115	. . . 4 |- ( A e. _V -> [_ A / x ]_ B = C )
6	2, 5	syl 14	. . 3 |- ( A e. V -> [_ A / x ]_ B = C )
7	6	csbeq1d 3079	. 2 |- ( A e. V -> [_ [_ A / x ]_ B / y ]_ D = [_ C / y ]_ D )
8	1, 7	eqtrd 2222	1 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ D )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160   _Vcvv 2752   [_csb 3072

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978  df-csb 3073

This theorem is referenced by: (None)