Theorem hbsb4t 1621

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1620). (The proof was shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
hbsb4t	|- (A.xA.z(ph -> A.zph) -> (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph)))

Proof of Theorem hbsb4t

Step	Hyp	Ref	Expression
1		a4sbim 1614	. . . . 5 |- (A.x(ph -> A.zph) -> ([y / x]ph -> [y / x]A.zph))
2	1	a4s 1330	. . . 4 |- (A.zA.x(ph -> A.zph) -> ([y / x]ph -> [y / x]A.zph))
3		ax-4 1319	. . . . . . 7 |- (A.zph -> ph)
4	3	sbimi 1537	. . . . . 6 |- ([y / x]A.zph -> [y / x]ph)
5	4	alimi 1338	. . . . 5 |- (A.z[y / x]A.zph -> A.z[y / x]ph)
6	5	a1i 8	. . . 4 |- (A.zA.x(ph -> A.zph) -> (A.z[y / x]A.zph -> A.z[y / x]ph))
7	2, 6	imim12d 69	. . 3 |- (A.zA.x(ph -> A.zph) -> (([y / x]A.zph -> A.z[y / x]A.zph) -> ([y / x]ph -> A.z[y / x]ph)))
8	7	a7s 1337	. 2 |- (A.xA.z(ph -> A.zph) -> (([y / x]A.zph -> A.z[y / x]A.zph) -> ([y / x]ph -> A.z[y / x]ph)))
9		hba1 1350	. . 3 |- (A.zph -> A.zA.zph)
10	9	hbsb4 1620	. 2 |- (-. A.z z = y -> ([y / x]A.zph -> A.z[y / x]A.zph))
11	8, 10	syl5 20	1 |- (A.xA.z(ph -> A.zph) -> (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph)))

Syntax hints:  -. wn 2   -> wi 3  A.wal 1296   = wceq 1298  [wsbc 1534

This theorem is referenced by:  dvelimdf 1624  hbabd 1876

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

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