Theorem nfsb4t 2065

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2063). (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 1590	. . . . 5 |- F/ z F/ z ph
2	1	nfal 1622	. . . 4 |- F/ z A. x F/ z ph
3		nfnae 1768	. . . 4 |- F/ z -. A. z z = y
4	2, 3	nfan 1611	. . 3 |- F/ z ( A. x F/ z ph /\ -. A. z z = y )
5		df-nf 1507	. . . . . 6 |- ( F/ z ph <-> A. z ( ph -> A. z ph ) )
6	5	albii 1516	. . . . 5 |- ( A. x F/ z ph <-> A. x A. z ( ph -> A. z ph ) )
7		hbsb4t 2064	. . . . 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 121	. . . 4 |- ( A. x F/ z ph -> ( -. A. z z = y -> ( [ y / x ] ph -> A. z [ y / x ] ph ) ) )
9	8	imp 124	. . 3 |- ( ( A. x F/ z ph /\ -. A. z z = y ) -> ( [ y / x ] ph -> A. z [ y / x ] ph ) )
10	4, 9	nfd 1569	. 2 |- ( ( A. x F/ z ph /\ -. A. z z = y ) -> F/ z [ y / x ] ph )
11	10	ex 115	1 |- ( A. x F/ z ph -> ( -. A. z z = y -> F/ z [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wal 1393   F/wnf 1506   [wsb 1808

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809

This theorem is referenced by:  dvelimdf  2067