Theorem nfsb4t 1989

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1987). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.)

Assertion

Ref	Expression
nfsb4t	|- ( A. x F/ z ph -> ( -. A. z z = y -> F/ z [ y / x ] ph ) )

Proof of Theorem nfsb4t

Step	Hyp	Ref	Expression
1		nfnf1 1523	. . . . 5 |- F/ z F/ z ph
2	1	nfal 1555	. . . 4 |- F/ z A. x F/ z ph
3		nfnae 1700	. . . 4 |- F/ z -. A. z z = y
4	2, 3	nfan 1544	. . 3 |- F/ z ( A. x F/ z ph /\ -. A. z z = y )
5		df-nf 1437	. . . . . 6 |- ( F/ z ph <-> A. z ( ph -> A. z ph ) )
6	5	albii 1446	. . . . 5 |- ( A. x F/ z ph <-> A. x A. z ( ph -> A. z ph ) )
7		hbsb4t 1988	. . . . 5 |- ( A. x A. z ( ph -> A. z ph ) -> ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) ) )
8	6, 7	sylbi 120	. . . 4 |- ( A. x F/ z ph -> ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) ) )
9	8	imp 123	. . 3 |- ( ( A. x F/ z ph /\ -. A. z z = y ) -> ( [ y / x ] ph -> A. z [ y / x ] ph ) )
10	4, 9	nfd 1503	. 2 |- ( ( A. x F/ z ph /\ -. A. z z = y ) -> F/ z [ y / x ] ph )
11	10	ex 114	1 |- ( A. x F/ z ph -> ( -. A. z z = y -> F/ z [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   A.wal 1329   F/wnf 1436   [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736

This theorem is referenced by:  dvelimdf  1991