Theorem csbco3g 3063

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 3059	. 2 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ [_ A / x ]_ B / y ]_ D )
2		elex 2700	. . . 4 |- ( A e. V -> A e. _V )
3		nfcvd 2283	. . . . 5 |- ( A e. _V -> F/_ x C )
4		sbcco3g.1	. . . . 5 |- ( x = A -> B = C )
5	3, 4	csbiegf 3048	. . . 4 |- ( A e. _V -> [_ A / x ]_ B = C )
6	2, 5	syl 14	. . 3 |- ( A e. V -> [_ A / x ]_ B = C )
7	6	csbeq1d 3014	. 2 |- ( A e. V -> [_ [_ A / x ]_ B / y ]_ D = [_ C / y ]_ D )
8	1, 7	eqtrd 2173	1 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ D )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   _Vcvv 2689   [_csb 3007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914  df-csb 3008

This theorem is referenced by: (None)