Theorem nfsb4t 2389

Description: A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2390). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-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		sbequ12 2111	. . . . . . . 8 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
2	1	sps 2055	. . . . . . 7 |- ( A. x x = y -> ( ph <-> [ y / x ] ph ) )
3	2	drnf2 2330	. . . . . 6 |- ( A. x x = y -> ( F/ z ph <-> F/ z [ y / x ] ph ) )
4	3	biimpd 219	. . . . 5 |- ( A. x x = y -> ( F/ z ph -> F/ z [ y / x ] ph ) )
5	4	spsd 2057	. . . 4 |- ( A. x x = y -> ( A. x F/ z ph -> F/ z [ y / x ] ph ) )
6	5	impcom 446	. . 3 |- ( ( A. x F/ z ph /\ A. x x = y ) -> F/ z [ y / x ] ph )
7	6	a1d 25	. 2 |- ( ( A. x F/ z ph /\ A. x x = y ) -> ( -. A. z z = y -> F/ z [ y / x ] ph ) )
8		nfnf1 2031	. . . . 5 |- F/ z F/ z ph
9	8	nfal 2153	. . . 4 |- F/ z A. x F/ z ph
10		nfnae 2318	. . . 4 |- F/ z -. A. x x = y
11	9, 10	nfan 1828	. . 3 |- F/ z ( A. x F/ z ph /\ -. A. x x = y )
12		nfa1 2028	. . . 4 |- F/ x A. x F/ z ph
13		nfnae 2318	. . . 4 |- F/ x -. A. x x = y
14	12, 13	nfan 1828	. . 3 |- F/ x ( A. x F/ z ph /\ -. A. x x = y )
15		sp 2053	. . . 4 |- ( A. x F/ z ph -> F/ z ph )
16	15	adantr 481	. . 3 |- ( ( A. x F/ z ph /\ -. A. x x = y ) -> F/ z ph )
17		nfsb2 2360	. . . 4 |- ( -. A. x x = y -> F/ x [ y / x ] ph )
18	17	adantl 482	. . 3 |- ( ( A. x F/ z ph /\ -. A. x x = y ) -> F/ x [ y / x ] ph )
19	1	a1i 11	. . 3 |- ( ( A. x F/ z ph /\ -. A. x x = y ) -> ( x = y -> ( ph <-> [ y / x ] ph ) ) )
20	11, 14, 16, 18, 19	dvelimdf 2335	. 2 |- ( ( A. x F/ z ph /\ -. A. x x = y ) -> ( -. A. z z = y -> F/ z [ y / x ] ph ) )
21	7, 20	pm2.61dan 832	1 |- ( A. x F/ z ph -> ( -. A. z z = y -> F/ z [ y / x ] ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   F/wnf 1708   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  nfsb4  2390  nfsbd  2442