Theorem sbcnestg 2955

Description: Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Assertion

Ref	Expression
sbcnestg	|- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )

Distinct variable group:   ph, x

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

Proof of Theorem sbcnestg

Step	Hyp	Ref	Expression
1		nfv 1461	. . 3 |- F/ x ph
2	1	ax-gen 1378	. 2 |- A. y F/ x ph
3		sbcnestgf 2953	. 2 |- ( ( A e. V /\ A. y F/ x ph ) -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
4	2, 3	mpan2 415	1 |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   F/wnf 1389   e. wcel 1433   [.wsbc 2815   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909

This theorem is referenced by:  sbcco3g  2959