Theorem hbsb4 1620

Description: A variable not free remains so after substitution with a distinct variable.

Hypothesis

Ref	Expression
hbsb4.1	|- (ph -> A.zph)

Assertion

Ref	Expression
hbsb4	|- (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph))

Proof of Theorem hbsb4

Step	Hyp	Ref	Expression
1		equequ1 1494	. . . . . 6 |- (z = x -> (z = y <-> x = y))
2	1	a4s 1330	. . . . 5 |- (A.z z = x -> (z = y <-> x = y))
3	2	dral1 1515	. . . 4 |- (A.z z = x -> (A.z z = y <-> A.x x = y))
4	3	notbid 673	. . 3 |- (A.z z = x -> (-. A.z z = y <-> -. A.x x = y))
5		hbsb2 1597	. . . 4 |- (-. A.x x = y -> ([y / x]ph -> A.x[y / x]ph))
6		ax-10o 1500	. . . . 5 |- (A.x x = z -> (A.x[y / x]ph -> A.z[y / x]ph))
7	6	alequcoms 1503	. . . 4 |- (A.z z = x -> (A.x[y / x]ph -> A.z[y / x]ph))
8	5, 7	syl9r 72	. . 3 |- (A.z z = x -> (-. A.x x = y -> ([y / x]ph -> A.z[y / x]ph)))
9	4, 8	sylbid 220	. 2 |- (A.z z = x -> (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph)))
10		hbae 1505	. . . . . 6 |- (A.x x = y -> A.zA.x x = y)
11		ax-4 1319	. . . . . . 7 |- (A.x x = y -> x = y)
12	11	alimi 1338	. . . . . 6 |- (A.zA.x x = y -> A.z x = y)
13		sbequ2 1543	. . . . . . . 8 |- (x = y -> ([y / x]ph -> ph))
14	13	a4s 1330	. . . . . . 7 |- (A.z x = y -> ([y / x]ph -> ph))
15		sbequ1 1542	. . . . . . . . 9 |- (x = y -> (ph -> [y / x]ph))
16	15	al2imi 1341	. . . . . . . 8 |- (A.z x = y -> (A.zph -> A.z[y / x]ph))
17		hbsb4.1	. . . . . . . 8 |- (ph -> A.zph)
18	16, 17	syl5 20	. . . . . . 7 |- (A.z x = y -> (ph -> A.z[y / x]ph))
19	14, 18	syld 30	. . . . . 6 |- (A.z x = y -> ([y / x]ph -> A.z[y / x]ph))
20	10, 12, 19	3syl 24	. . . . 5 |- (A.x x = y -> ([y / x]ph -> A.z[y / x]ph))
21	20	a1d 15	. . . 4 |- (A.x x = y -> ((-. A.z z = x /\ -. A.z z = y) -> ([y / x]ph -> A.z[y / x]ph)))
22		sb4 1593	. . . . 5 |- (-. A.x x = y -> ([y / x]ph -> A.x(x = y -> ph)))
23		hbnae 1507	. . . . . . . 8 |- (-. A.z z = x -> A.x -. A.z z = x)
24		hbnae 1507	. . . . . . . 8 |- (-. A.z z = y -> A.x -. A.z z = y)
25	23, 24	hban 1356	. . . . . . 7 |- ((-. A.z z = x /\ -. A.z z = y) -> A.x(-. A.z z = x /\ -. A.z z = y))
26		hbnae 1507	. . . . . . . . 9 |- (-. A.z z = x -> A.z -. A.z z = x)
27		hbnae 1507	. . . . . . . . 9 |- (-. A.z z = y -> A.z -. A.z z = y)
28	26, 27	hban 1356	. . . . . . . 8 |- ((-. A.z z = x /\ -. A.z z = y) -> A.z(-. A.z z = x /\ -. A.z z = y))
29		ax-12 1310	. . . . . . . . 9 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))
30	29	imp 377	. . . . . . . 8 |- ((-. A.z z = x /\ -. A.z z = y) -> (x = y -> A.z x = y))
31	17	a1i 8	. . . . . . . 8 |- ((-. A.z z = x /\ -. A.z z = y) -> (ph -> A.zph))
32	28, 30, 31	hbimd 1468	. . . . . . 7 |- ((-. A.z z = x /\ -. A.z z = y) -> ((x = y -> ph) -> A.z(x = y -> ph)))
33	25, 32	alimd 1343	. . . . . 6 |- ((-. A.z z = x /\ -. A.z z = y) -> (A.x(x = y -> ph) -> A.xA.z(x = y -> ph)))
34		sb2 1541	. . . . . . . 8 |- (A.x(x = y -> ph) -> [y / x]ph)
35	34	alimi 1338	. . . . . . 7 |- (A.zA.x(x = y -> ph) -> A.z[y / x]ph)
36	35	a7s 1337	. . . . . 6 |- (A.xA.z(x = y -> ph) -> A.z[y / x]ph)
37	33, 36	syl6 25	. . . . 5 |- ((-. A.z z = x /\ -. A.z z = y) -> (A.x(x = y -> ph) -> A.z[y / x]ph))
38	22, 37	syl9 71	. . . 4 |- (-. A.x x = y -> ((-. A.z z = x /\ -. A.z z = y) -> ([y / x]ph -> A.z[y / x]ph)))
39	21, 38	pm2.61i 140	. . 3 |- ((-. A.z z = x /\ -. A.z z = y) -> ([y / x]ph -> A.z[y / x]ph))
40	39	ex 402	. 2 |- (-. A.z z = x -> (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph)))
41	9, 40	pm2.61i 140	1 |- (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298  [wsbc 1534

This theorem is referenced by:  hbsb4t 1621  hbsb4tOLD 1622  dvelimf 1623  sbco2 1629  hbsb 1723  sbal1 1737  hbab 1875

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536

Copyright terms: Public domain