Theorem csbnestg 3202

Description: Nest the composition of two substitutions. (Contributed by NM, 23-Nov-2005.) (Proof shortened by Mario Carneiro, 10-Nov-2016.)

Assertion

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

Distinct variable group:   x, C

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

Proof of Theorem csbnestg

Step	Hyp	Ref	Expression
1		nfcv 2392	. . 3 |- F/_ x C
2	1	ax-gen 1502	. 2 |- A. y F/_ x C
3		csbnestgf 3200	. 2 |- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )
4	2, 3	mpan2 429	1 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )

Syntax hints:   -> wi 4   A.wal 1400   = wceq 1402   e. wcel 2209   F/_wnfc 2379   [_csb 3147

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052  df-csb 3148

This theorem is referenced by:  csbco3g  3206