Theorem sbcnestg 3155

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 1552	. . 3 |- F/ x ph
2	1	ax-gen 1473	. 2 |- A. y F/ x ph
3		sbcnestgf 3153	. 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 1371   F/wnf 1484   e. wcel 2178   [.wsbc 3005   [_csb 3101

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-sbc 3006  df-csb 3102

This theorem is referenced by:  sbcco3g  3159