Theorem nfsb4t 1950

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1948). (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 1491	. . . . 5 |- F/ z F/ z ph
2	1	nfal 1523	. . . 4 |- F/ z A. x F/ z ph
3		nfnae 1668	. . . 4 |- F/ z -. A. z z = y
4	2, 3	nfan 1512	. . 3 |- F/ z ( A. x F/ z ph /\ -. A. z z = y )
5		df-nf 1405	. . . . . 6 |- ( F/ z ph <-> A. z ( ph -> A. z ph ) )
6	5	albii 1414	. . . . 5 |- ( A. x F/ z ph <-> A. x A. z ( ph -> A. z ph ) )
7		hbsb4t 1949	. . . . 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 1471	. 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 1297   F/wnf 1404   [wsb 1703

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704

This theorem is referenced by:  dvelimdf  1952