Theorem sbcnestg 3182

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 1577	. . 3 |- F/ x ph
2	1	ax-gen 1498	. 2 |- A. y F/ x ph
3		sbcnestgf 3180	. 2 |- ( ( A e. V /\ A. y F/ x ph ) -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
4	2, 3	mpan2 425	1 |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1396   F/wnf 1509   e. wcel 2202   [.wsbc 3032   [_csb 3128

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-sbc 3033  df-csb 3129

This theorem is referenced by:  sbcco3g  3186