Theorem ralxfr 3839

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

Hypotheses

Ref	Expression
ralxfr.1	|- (y e. B -> A e. B)
ralxfr.2	|- (x e. B -> E.y e. B x = A)
ralxfr.3	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
ralxfr	|- (A.x e. B ph <-> A.y e. B ps)

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

Proof of Theorem ralxfr

Step	Hyp	Ref	Expression
1		ralxfr.1	. . . . 5 |- (y e. B -> A e. B)
2		ralxfr.3	. . . . . 6 |- (x = A -> (ph <-> ps))
3	2	rcla4v 2376	. . . . 5 |- (A e. B -> (A.x e. B ph -> ps))
4	1, 3	syl 12	. . . 4 |- (y e. B -> (A.x e. B ph -> ps))
5	4	com12 14	. . 3 |- (A.x e. B ph -> (y e. B -> ps))
6	5	r19.21aiv 2175	. 2 |- (A.x e. B ph -> A.y e. B ps)
7		hbra1 2147	. . . . 5 |- (A.y e. B ps -> A.yA.y e. B ps)
8		ax-17 1317	. . . . 5 |- (ph -> A.yph)
9		ra4 2155	. . . . . 6 |- (A.y e. B ps -> (y e. B -> ps))
10	2	biimprcd 173	. . . . . 6 |- (ps -> (x = A -> ph))
11	9, 10	syl6 25	. . . . 5 |- (A.y e. B ps -> (y e. B -> (x = A -> ph)))
12	7, 8, 11	r19.23ad 2213	. . . 4 |- (A.y e. B ps -> (E.y e. B x = A -> ph))
13		ralxfr.2	. . . 4 |- (x e. B -> E.y e. B x = A)
14	12, 13	syl5 20	. . 3 |- (A.y e. B ps -> (x e. B -> ph))
15	14	r19.21aiv 2175	. 2 |- (A.y e. B ps -> A.x e. B ph)
16	6, 15	impbii 174	1 |- (A.x e. B ph <-> A.y e. B ps)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106

This theorem is referenced by:  rexxfr 3841  infm3 7263  infmsup 7277

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-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294

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