Theorem rexxfrd 2898

Description: Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by FL, 10-Apr-2007.)

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
rexxfrd	|- (ph -> (E.x e. B ps <-> E.y e. B ch))

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

Proof of Theorem rexxfrd

Step	Hyp	Ref	Expression
1		ralxfrd.1	. . . 4 |- ((ph /\ y e. B) -> A e. B)
2		ralxfrd.2	. . . 4 |- ((ph /\ x e. B) -> E.y e. B x = A)
3		ralxfrd.3	. . . . 5 |- ((ph /\ x = A) -> (ps <-> ch))
4	3	negbid 611	. . . 4 |- ((ph /\ x = A) -> (-. ps <-> -. ch))
5	1, 2, 4	ralxfrd 2897	. . 3 |- (ph -> (A.x e. B -. ps <-> A.y e. B -. ch))
6	5	negbid 611	. 2 |- (ph -> (-. A.x e. B -. ps <-> -. A.y e. B -. ch))
7		dfrex2 1656	. 2 |- (E.x e. B ps <-> -. A.x e. B -. ps)
8		dfrex2 1656	. 2 |- (E.y e. B ch <-> -. A.y e. B -. ch)
9	6, 7, 8	3bitr4g 555	1 |- (ph -> (E.x e. B ps <-> E.y e. B ch))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646

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

Copyright terms: Public domain