Theorem csbnestg 2900

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 2178	. . 3 |- F/_ x C
2	1	ax-gen 1338	. 2 |- A. y F/_ x C
3		csbnestgf 2898	. 2 |- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )
4	2, 3	mpan2 401	1 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )

Syntax hints:   -> wi 4   A.wal 1241   = wceq 1243   e. wcel 1393   F/_wnfc 2165   [_csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by:  csbco3g  2904