Theorem csbnestg 3103

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 2312	. . 3 |- F/_ x C
2	1	ax-gen 1442	. 2 |- A. y F/_ x C
3		csbnestgf 3101	. 2 |- ( ( A e. V /\ A. y F/_ x C ) -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )
4	2, 3	mpan2 423	1 |- ( A e. V -> [_ A / x ]_ [_ B / y ]_ C = [_ [_ A / x ]_ B / y ]_ C )

Syntax hints:   -> wi 4   A.wal 1346   = wceq 1348   e. wcel 2141   F/_wnfc 2299   [_csb 3049

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by:  csbco3g  3107