Theorem nfsb4t 2042

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 2040). (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 1567	. . . . 5 |- F/ z F/ z ph
2	1	nfal 1599	. . . 4 |- F/ z A. x F/ z ph
3		nfnae 1745	. . . 4 |- F/ z -. A. z z = y
4	2, 3	nfan 1588	. . 3 |- F/ z ( A. x F/ z ph /\ -. A. z z = y )
5		df-nf 1484	. . . . . 6 |- ( F/ z ph <-> A. z ( ph -> A. z ph ) )
6	5	albii 1493	. . . . 5 |- ( A. x F/ z ph <-> A. x A. z ( ph -> A. z ph ) )
7		hbsb4t 2041	. . . . 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 1546	. 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 1371   F/wnf 1483   [wsb 1785

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786

This theorem is referenced by:  dvelimdf  2044