Theorem sbcnestg 3146

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 1550	. . 3 |- F/ x ph
2	1	ax-gen 1471	. 2 |- A. y F/ x ph
3		sbcnestgf 3144	. 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 1370   F/wnf 1482   e. wcel 2175   [.wsbc 2997   [_csb 3092

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-sbc 2998  df-csb 3093

This theorem is referenced by:  sbcco3g  3150