Theorem sbcnestg 3186

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 1619	. . 3 |- F/xph
2	1	ax-gen 1546	. 2 |- A.y F/xph
3		sbcnestgf 3184	. 2 |- ((A e. V /\ A.y F/xph) -> ( [.A / x ]. [.B / y ].ph <-> [.[_A / x]_B / y ].ph))
4	2, 3	mpan2 652	1 |- (A e. V -> ( [.A / x ]. [.B / y ].ph <-> [.[_A / x]_B / y ].ph))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   F/wnf 1544   e. wcel 1710   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  sbcco3g  3192