Theorem alxfr 2886

Description: Transfer universal quantification from a variable x to another variable y contained in expression A.

Hypothesis

Ref	Expression
alxfr.1	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
alxfr	|- ((A.y A e. B /\ A.xE.y x = A) -> (A.xph <-> A.yps))

Distinct variable groups:   x,A   ph,y   ps,x   x,y

Proof of Theorem alxfr

Step	Hyp	Ref	Expression
1		hba1 1000	. . . 4 |- (A.y A e. B -> A.yA.y A e. B)
2		ax-17 968	. . . 4 |- (A.xph -> A.yA.xph)
3		alxfr.1	. . . . . 6 |- (x = A -> (ph <-> ps))
4	3	cla4gv 1853	. . . . 5 |- (A e. B -> (A.xph -> ps))
5	4	a4s 981	. . . 4 |- (A.y A e. B -> (A.xph -> ps))
6	1, 2, 5	19.21ad 1055	. . 3 |- (A.y A e. B -> (A.xph -> A.yps))
7	6	adantr 389	. 2 |- ((A.y A e. B /\ A.xE.y x = A) -> (A.xph -> A.yps))
8		hba1 1000	. . . 4 |- (A.xE.y x = A -> A.xA.xE.y x = A)
9		ax-17 968	. . . 4 |- (A.yps -> A.xA.yps)
10		hba1 1000	. . . . . . 7 |- (A.yps -> A.yA.yps)
11		ax-17 968	. . . . . . 7 |- (ph -> A.yph)
12	10, 11	hbim 1004	. . . . . 6 |- ((A.yps -> ph) -> A.y(A.yps -> ph))
13		ax-4 970	. . . . . . 7 |- (A.yps -> ps)
14	3, 13	syl5bir 210	. . . . . 6 |- (x = A -> (A.yps -> ph))
15	12, 14	19.23ai 1060	. . . . 5 |- (E.y x = A -> (A.yps -> ph))
16	15	a4s 981	. . . 4 |- (A.xE.y x = A -> (A.yps -> ph))
17	8, 9, 16	19.21ad 1055	. . 3 |- (A.xE.y x = A -> (A.yps -> A.xph))
18	17	adantl 388	. 2 |- ((A.y A e. B /\ A.xE.y x = A) -> (A.yps -> A.xph))
19	7, 18	impbid 514	1 |- ((A.y A e. B /\ A.xE.y x = A) -> (A.xph <-> A.yps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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