Theorem ralxfrd 3837

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

Hypotheses

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

Assertion

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

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

Proof of Theorem ralxfrd

Step	Hyp	Ref	Expression
1		ralxfrd.1	. . . . . 6 |- ((ph /\ y e. B) -> A e. B)
2	1	ex 402	. . . . 5 |- (ph -> (y e. B -> A e. B))
3		ralxfrd.3	. . . . . . 7 |- ((ph /\ x = A) -> (ps <-> ch))
4	3	rcla4dv 2382	. . . . . 6 |- ((ph /\ A e. B) -> (A.x e. B ps -> ch))
5	4	ex 402	. . . . 5 |- (ph -> (A e. B -> (A.x e. B ps -> ch)))
6	2, 5	syld 30	. . . 4 |- (ph -> (y e. B -> (A.x e. B ps -> ch)))
7	6	com23 36	. . 3 |- (ph -> (A.x e. B ps -> (y e. B -> ch)))
8	7	r19.21adv 2181	. 2 |- (ph -> (A.x e. B ps -> A.y e. B ch))
9		ax-17 1317	. . . . . . 7 |- (ph -> A.yph)
10		hbra1 2147	. . . . . . 7 |- (A.y e. B ch -> A.yA.y e. B ch)
11	9, 10	hban 1356	. . . . . 6 |- ((ph /\ A.y e. B ch) -> A.y(ph /\ A.y e. B ch))
12		ax-17 1317	. . . . . 6 |- (ps -> A.yps)
13		ra4 2155	. . . . . . 7 |- (A.y e. B ch -> (y e. B -> ch))
14	3	exbiri 421	. . . . . . . 8 |- (ph -> (x = A -> (ch -> ps)))
15	14	com23 36	. . . . . . 7 |- (ph -> (ch -> (x = A -> ps)))
16	13, 15	sylan9r 519	. . . . . 6 |- ((ph /\ A.y e. B ch) -> (y e. B -> (x = A -> ps)))
17	11, 12, 16	r19.23ad 2213	. . . . 5 |- ((ph /\ A.y e. B ch) -> (E.y e. B x = A -> ps))
18	17	ex 402	. . . 4 |- (ph -> (A.y e. B ch -> (E.y e. B x = A -> ps)))
19		ralxfrd.2	. . . . 5 |- ((ph /\ x e. B) -> E.y e. B x = A)
20	19	ex 402	. . . 4 |- (ph -> (x e. B -> E.y e. B x = A))
21	18, 20	syl5d 66	. . 3 |- (ph -> (A.y e. B ch -> (x e. B -> ps)))
22	21	r19.21adv 2181	. 2 |- (ph -> (A.y e. B ch -> A.x e. B ps))
23	8, 22	impbid 574	1 |- (ph -> (A.x e. B ps <-> A.y e. B ch))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106

This theorem is referenced by:  rexxfrd 3838  ralxfrALT 3840  islp2 9023  glbcon 17028

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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