Theorem rexxfr 2917

Description: Transfer existence 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
rexxfr	|- (E.x e. B ph <-> E.y e. B ps)

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

Proof of Theorem rexxfr

Step	Hyp	Ref	Expression
1		ralxfr.1	. . . 4 |- (y e. B -> A e. B)
2		ralxfr.2	. . . 4 |- (x e. B -> E.y e. B x = A)
3		ralxfr.3	. . . . 5 |- (x = A -> (ph <-> ps))
4	3	notbid 614	. . . 4 |- (x = A -> (-. ph <-> -. ps))
5	1, 2, 4	ralxfr 2915	. . 3 |- (A.x e. B -. ph <-> A.y e. B -. ps)
6	5	notbii 187	. 2 |- (-. A.x e. B -. ph <-> -. A.y e. B -. ps)
7		dfrex2 1663	. 2 |- (E.x e. B ph <-> -. A.x e. B -. ph)
8		dfrex2 1663	. 2 |- (E.y e. B ps <-> -. A.y e. B -. ps)
9	6, 7, 8	3bitr4i 183	1 |- (E.x e. B ph <-> E.y e. B ps)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 960   e. wcel 962  A.wral 1652  E.wrex 1653

This theorem is referenced by:  infm3 6086  infmsup 6100  reeff1o 7458

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-ral 1656  df-rex 1657  df-v 1819

Copyright terms: Public domain