Theorem cbvexh 1685

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. y ph )
cbvexh.2	|- ( ps -> A. x ps )
cbvexh.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvexh	|- ( E. x ph <-> E. y ps )

Proof of Theorem cbvexh

Step	Hyp	Ref	Expression
1		cbvexh.2	. . . 4 |- ( ps -> A. x ps )
2	1	hbex 1572	. . 3 |- ( E. y ps -> A. x E. y ps )
3		cbvexh.1	. . . . 5 |- ( ph -> A. y ph )
4		cbvexh.3	. . . . . . 7 |- ( x = y -> ( ph <-> ps ) )
5	4	bicomd 139	. . . . . 6 |- ( x = y -> ( ps <-> ph ) )
6	5	equcoms 1641	. . . . 5 |- ( y = x -> ( ps <-> ph ) )
7	3, 6	equsex 1663	. . . 4 |- ( E. y ( y = x /\ ps ) <-> ph )
8		simpr 108	. . . . 5 |- ( ( y = x /\ ps ) -> ps )
9	8	eximi 1536	. . . 4 |- ( E. y ( y = x /\ ps ) -> E. y ps )
10	7, 9	sylbir 133	. . 3 |- ( ph -> E. y ps )
11	2, 10	exlimih 1529	. 2 |- ( E. x ph -> E. y ps )
12	3	hbex 1572	. . 3 |- ( E. x ph -> A. y E. x ph )
13	1, 4	equsex 1663	. . . 4 |- ( E. x ( x = y /\ ph ) <-> ps )
14		simpr 108	. . . . 5 |- ( ( x = y /\ ph ) -> ph )
15	14	eximi 1536	. . . 4 |- ( E. x ( x = y /\ ph ) -> E. x ph )
16	13, 15	sylbir 133	. . 3 |- ( ps -> E. x ph )
17	12, 16	exlimih 1529	. 2 |- ( E. y ps -> E. x ph )
18	11, 17	impbii 124	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1287   = wceq 1289   E.wex 1426

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  cbvex  1686  sb8eh  1783  cbvexv  1843  euf  1953  mopick  2026