Theorem csbco3g 3152

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 3148	. 2 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ [_ A / x ]_ B / y ]_ D )
2		elex 2783	. . . 4 |- ( A e. V -> A e. _V )
3		nfcvd 2349	. . . . 5 |- ( A e. _V -> F/_ x C )
4		sbcco3g.1	. . . . 5 |- ( x = A -> B = C )
5	3, 4	csbiegf 3137	. . . 4 |- ( A e. _V -> [_ A / x ]_ B = C )
6	2, 5	syl 14	. . 3 |- ( A e. V -> [_ A / x ]_ B = C )
7	6	csbeq1d 3100	. 2 |- ( A e. V -> [_ [_ A / x ]_ B / y ]_ D = [_ C / y ]_ D )
8	1, 7	eqtrd 2238	1 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ D = [_ C / y ]_ D )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   _Vcvv 2772   [_csb 3093

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999  df-csb 3094

This theorem is referenced by: (None)