Theorem nfsb4t 2080

Description: A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2081). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.)

Assertion

Ref	Expression
nfsb4t	|- (A.x F/zph -> (-. A.z z = y -> F/z[y / x]ph))

Proof of Theorem nfsb4t

Step	Hyp	Ref	Expression
1		sbequ12 1919	. . . . . . . . 9 |- (x = y -> (ph <-> [y / x]ph))
2	1	sps 1754	. . . . . . . 8 |- (A.x x = y -> (ph <-> [y / x]ph))
3	2	drnf2 1970	. . . . . . 7 |- (A.x x = y -> ( F/zph <-> F/z[y / x]ph))
4	3	biimpcd 215	. . . . . 6 |- ( F/zph -> (A.x x = y -> F/z[y / x]ph))
5	4	sps 1754	. . . . 5 |- (A.x F/zph -> (A.x x = y -> F/z[y / x]ph))
6	5	a1dd 42	. . . 4 |- (A.x F/zph -> (A.x x = y -> ((-. A.z z = x /\ -. A.z z = y) -> F/z[y / x]ph)))
7		nfa1 1788	. . . . . . . 8 |- F/xA.x F/zph
8		nfnae 1956	. . . . . . . . 9 |- F/x -. A.z z = x
9		nfnae 1956	. . . . . . . . 9 |- F/x -. A.z z = y
10	8, 9	nfan 1824	. . . . . . . 8 |- F/x(-. A.z z = x /\ -. A.z z = y)
11	7, 10	nfan 1824	. . . . . . 7 |- F/x(A.x F/zph /\ (-. A.z z = x /\ -. A.z z = y))
12		nfeqf 1958	. . . . . . . . 9 |- ((-. A.z z = x /\ -. A.z z = y) -> F/z x = y)
13	12	adantl 452	. . . . . . . 8 |- ((A.x F/zph /\ (-. A.z z = x /\ -. A.z z = y)) -> F/z x = y)
14		sp 1747	. . . . . . . . 9 |- (A.x F/zph -> F/zph)
15	14	adantr 451	. . . . . . . 8 |- ((A.x F/zph /\ (-. A.z z = x /\ -. A.z z = y)) -> F/zph)
16	13, 15	nfimd 1808	. . . . . . 7 |- ((A.x F/zph /\ (-. A.z z = x /\ -. A.z z = y)) -> F/z(x = y -> ph))
17	11, 16	nfald 1852	. . . . . 6 |- ((A.x F/zph /\ (-. A.z z = x /\ -. A.z z = y)) -> F/zA.x(x = y -> ph))
18	17	ex 423	. . . . 5 |- (A.x F/zph -> ((-. A.z z = x /\ -. A.z z = y) -> F/zA.x(x = y -> ph)))
19		nfnae 1956	. . . . . . 7 |- F/z -. A.x x = y
20		sb4b 2054	. . . . . . 7 |- (-. A.x x = y -> ([y / x]ph <-> A.x(x = y -> ph)))
21	19, 20	nfbidf 1774	. . . . . 6 |- (-. A.x x = y -> ( F/z[y / x]ph <-> F/zA.x(x = y -> ph)))
22	21	imbi2d 307	. . . . 5 |- (-. A.x x = y -> (((-. A.z z = x /\ -. A.z z = y) -> F/z[y / x]ph) <-> ((-. A.z z = x /\ -. A.z z = y) -> F/zA.x(x = y -> ph))))
23	18, 22	syl5ibrcom 213	. . . 4 |- (A.x F/zph -> (-. A.x x = y -> ((-. A.z z = x /\ -. A.z z = y) -> F/z[y / x]ph)))
24	6, 23	pm2.61d 150	. . 3 |- (A.x F/zph -> ((-. A.z z = x /\ -. A.z z = y) -> F/z[y / x]ph))
25	24	exp3a 425	. 2 |- (A.x F/zph -> (-. A.z z = x -> (-. A.z z = y -> F/z[y / x]ph)))
26		nfsb2 2058	. . 3 |- (-. A.z z = y -> F/z[y / z]ph)
27		drsb1 2022	. . . 4 |- (A.z z = x -> ([y / z]ph <-> [y / x]ph))
28	27	drnf2 1970	. . 3 |- (A.z z = x -> ( F/z[y / z]ph <-> F/z[y / x]ph))
29	26, 28	syl5ib 210	. 2 |- (A.z z = x -> (-. A.z z = y -> F/z[y / x]ph))
30	25, 29	pm2.61d2 152	1 |- (A.x F/zph -> (-. A.z z = y -> F/z[y / x]ph))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  nfsb4  2081  dvelimdf  2082  nfsbd  2111