Theorem ralxfrd 2897

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 373	. . . . 5 |- (ph -> (y e. B -> A e. B))
3		ralxfrd.3	. . . . . . 7 |- ((ph /\ x = A) -> (ps <-> ch))
4	3	rcla4dv 1878	. . . . . 6 |- ((ph /\ A e. B) -> (A.x e. B ps -> ch))
5	4	ex 373	. . . . 5 |- (ph -> (A e. B -> (A.x e. B ps -> ch)))
6	2, 5	syld 27	. . . 4 |- (ph -> (y e. B -> (A.x e. B ps -> ch)))
7	6	com23 32	. . 3 |- (ph -> (A.x e. B ps -> (y e. B -> ch)))
8	7	r19.21adv 1718	. 2 |- (ph -> (A.x e. B ps -> A.y e. B ch))
9		ax-17 971	. . . . . . 7 |- (ph -> A.yph)
10		hbra1 1687	. . . . . . 7 |- (A.y e. B ch -> A.yA.y e. B ch)
11	9, 10	hban 1009	. . . . . 6 |- ((ph /\ A.y e. B ch) -> A.y(ph /\ A.y e. B ch))
12		ax-17 971	. . . . . 6 |- (ps -> A.yps)
13		ra4 1694	. . . . . . 7 |- (A.y e. B ch -> (y e. B -> ch))
14	3	biimprd 154	. . . . . . . . 9 |- ((ph /\ x = A) -> (ch -> ps))
15	14	ex 373	. . . . . . . 8 |- (ph -> (x = A -> (ch -> ps)))
16	15	com23 32	. . . . . . 7 |- (ph -> (ch -> (x = A -> ps)))
17	13, 16	sylan9r 469	. . . . . 6 |- ((ph /\ A.y e. B ch) -> (y e. B -> (x = A -> ps)))
18	11, 12, 17	r19.23ad 1745	. . . . 5 |- ((ph /\ A.y e. B ch) -> (E.y e. B x = A -> ps))
19	18	ex 373	. . . 4 |- (ph -> (A.y e. B ch -> (E.y e. B x = A -> ps)))
20		ralxfrd.2	. . . . 5 |- ((ph /\ x e. B) -> E.y e. B x = A)
21	20	ex 373	. . . 4 |- (ph -> (x e. B -> E.y e. B x = A))
22	19, 21	syl5d 55	. . 3 |- (ph -> (A.y e. B ch -> (x e. B -> ps)))
23	22	r19.21adv 1718	. 2 |- (ph -> (A.y e. B ch -> A.x e. B ps))
24	8, 23	impbid 516	1 |- (ph -> (A.x e. B ps <-> A.y e. B ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646

This theorem is referenced by:  rexxfrd 2898  ralxfrALT 2900  islp2 7747

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812

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