Theorem hbsb4t 906

Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 905).

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		ax-4 673	. . . . . 6 |- (A.zph -> ph)
2	1	biantru 543	. . . . 5 |- ((ph -> A.zph) <-> ((ph -> A.zph) /\ (A.zph -> ph)))
3		bi 396	. . . . 5 |- ((ph <-> A.zph) <-> ((ph -> A.zph) /\ (A.zph -> ph)))
4	2, 3	bitr4 154	. . . 4 |- ((ph -> A.zph) <-> (ph <-> A.zph))
5	4	bi2al 696	. . 3 |- (A.xA.z(ph -> A.zph) <-> A.xA.z(ph <-> A.zph))
6		sbba4 896	. . . . . 6 |- (A.x(ph <-> A.zph) -> ([y / x]ph <-> [y / x]A.zph))
7	6	a4s 682	. . . . 5 |- (A.zA.x(ph <-> A.zph) -> ([y / x]ph <-> [y / x]A.zph))
8		hba1 698	. . . . . 6 |- (A.zA.x(ph <-> A.zph) -> A.zA.zA.x(ph <-> A.zph))
9	8, 7	biald 782	. . . . 5 |- (A.zA.x(ph <-> A.zph) -> (A.z[y / x]ph <-> A.z[y / x]A.zph))
10	7, 9	imbi12d 474	. . . 4 |- (A.zA.x(ph <-> A.zph) -> (([y / x]ph -> A.z[y / x]ph) <-> ([y / x]A.zph -> A.z[y / x]A.zph)))
11	10	a7s 689	. . 3 |- (A.xA.z(ph <-> A.zph) -> (([y / x]ph -> A.z[y / x]ph) <-> ([y / x]A.zph -> A.z[y / x]A.zph)))
12	5, 11	sylbi 174	. 2 |- (A.xA.z(ph -> A.zph) -> (([y / x]ph -> A.z[y / x]ph) <-> ([y / x]A.zph -> A.z[y / x]A.zph)))
13		hba1 698	. . 3 |- (A.zph -> A.zA.zph)
14	13	hbsb4 905	. 2 |- (-. A.z z = y -> ([y / x]A.zph -> A.z[y / x]A.zph))
15	12, 14	syl5bir 184	1 |- (A.xA.z(ph -> A.zph) -> (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   = weq 797  [wsb 852

This theorem is referenced by:  ddelimdf 909

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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