Theorem nfsb4t 1890

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1888). (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 1436	. . . . 5 |- F/ z F/ z ph
2	1	nfal 1468	. . . 4 |- F/ z A. x F/ z ph
3		nfnae 1610	. . . 4 |- F/ z -. A. z z = y
4	2, 3	nfan 1457	. . 3 |- F/ z ( A. x F/ z ph /\ -. A. z z = y )
5		df-nf 1350	. . . . . 6 |- ( F/ z ph <-> A. z ( ph -> A. z ph ) )
6	5	albii 1359	. . . . 5 |- ( A. x F/ z ph <-> A. x A. z ( ph -> A. z ph ) )
7		hbsb4t 1889	. . . . 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 114	. . . 4 |- ( A. x F/ z ph -> ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) ) )
9	8	imp 115	. . 3 |- ( ( A. x F/ z ph /\ -. A. z z = y ) -> ( [ y / x ] ph -> A. z [ y / x ] ph ) )
10	4, 9	nfd 1416	. 2 |- ( ( A. x F/ z ph /\ -. A. z z = y ) -> F/ z [ y / x ] ph )
11	10	ex 108	1 |- ( A. x F/ z ph -> ( -. A. z z = y -> F/ z [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   A.wal 1241   F/wnf 1349   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646

This theorem is referenced by:  dvelimdf  1892