Theorem cbvexh 1635

Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Feb-2015.)

Hypotheses

Ref	Expression
cbvexh.1	|- (ph -> A.yph)
cbvexh.2	|- (ps -> A.xps)
cbvexh.3	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvexh	|- (E.xph <-> E.yps)

Proof of Theorem cbvexh

Step	Hyp	Ref	Expression
1		cbvexh.2	. . . 4 |- (ps -> A.xps)
2	1	hbex 1524	. . 3 |- (E.yps -> A.xE.yps)
3		cbvexh.1	. . . . 5 |- (ph -> A.yph)
4		cbvexh.3	. . . . . . 7 |- (x = y -> (ph <-> ps))
5	4	bicomd 129	. . . . . 6 |- (x = y -> (ps <-> ph))
6	5	equcoms 1591	. . . . 5 |- (y = x -> (ps <-> ph))
7	3, 6	equsex 1613	. . . 4 |- (E.y(y = x /\ ps) <-> ph)
8		simpr 103	. . . . 5 |- ((y = x /\ ps) -> ps)
9	8	eximi 1488	. . . 4 |- (E.y(y = x /\ ps) -> E.yps)
10	7, 9	sylbir 125	. . 3 |- (ph -> E.yps)
11	2, 10	exlimih 1481	. 2 |- (E.xph -> E.yps)
12	3	hbex 1524	. . 3 |- (E.xph -> A.yE.xph)
13	1, 4	equsex 1613	. . . 4 |- (E.x(x = y /\ ph) <-> ps)
14		simpr 103	. . . . 5 |- ((x = y /\ ph) -> ph)
15	14	eximi 1488	. . . 4 |- (E.x(x = y /\ ph) -> E.xph)
16	13, 15	sylbir 125	. . 3 |- (ps -> E.xph)
17	12, 16	exlimih 1481	. 2 |- (E.yps -> E.xph)
18	11, 17	impbii 117	1 |- (E.xph <-> E.yps)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   = wceq 1242  E.wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  cbvex  1636  sb8eh  1732  cbvexv  1792  euf  1902  mopick  1975