Theorem cbvexdh 1861

Description: Deduction used to change bound variables, using implicit substitition, particularly useful in conjunction with dvelim 1953. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.)

Hypotheses

Ref	Expression
cbvexdh.1	|- ( ph -> A. y ph )
cbvexdh.2	|- ( ph -> ( ps -> A. y ps ) )
cbvexdh.3	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

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

Distinct variable groups:   ph, x   ch, x

Allowed substitution hints:   ph( y)   ps( x, y)   ch( y)

Proof of Theorem cbvexdh

Step	Hyp	Ref	Expression
1		ax-17 1474	. . 3 |- ( ph -> A. x ph )
2		ax-17 1474	. . . 4 |- ( ch -> A. x ch )
3	2	hbex 1583	. . 3 |- ( E. y ch -> A. x E. y ch )
4		cbvexdh.1	. . . . 5 |- ( ph -> A. y ph )
5		cbvexdh.2	. . . . 5 |- ( ph -> ( ps -> A. y ps ) )
6		cbvexdh.3	. . . . . 6 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
7		equcomi 1648	. . . . . . 7 |- ( y = x -> x = y )
8		bicom1 130	. . . . . . 7 |- ( ( ps <-> ch ) -> ( ch <-> ps ) )
9	7, 8	imim12i 59	. . . . . 6 |- ( ( x = y -> ( ps <-> ch ) ) -> ( y = x -> ( ch <-> ps ) ) )
10	6, 9	syl 14	. . . . 5 |- ( ph -> ( y = x -> ( ch <-> ps ) ) )
11	4, 5, 10	equsexd 1675	. . . 4 |- ( ph -> ( E. y ( y = x /\ ch ) <-> ps ) )
12		simpr 109	. . . . 5 |- ( ( y = x /\ ch ) -> ch )
13	12	eximi 1547	. . . 4 |- ( E. y ( y = x /\ ch ) -> E. y ch )
14	11, 13	syl6bir 163	. . 3 |- ( ph -> ( ps -> E. y ch ) )
15	1, 3, 14	exlimdh 1543	. 2 |- ( ph -> ( E. x ps -> E. y ch ) )
16	1, 5	eximdh 1558	. . . 4 |- ( ph -> ( E. x ps -> E. x A. y ps ) )
17		19.12 1611	. . . 4 |- ( E. x A. y ps -> A. y E. x ps )
18	16, 17	syl6 33	. . 3 |- ( ph -> ( E. x ps -> A. y E. x ps ) )
19	2	a1i 9	. . . . 5 |- ( ph -> ( ch -> A. x ch ) )
20	1, 19, 6	equsexd 1675	. . . 4 |- ( ph -> ( E. x ( x = y /\ ps ) <-> ch ) )
21		simpr 109	. . . . 5 |- ( ( x = y /\ ps ) -> ps )
22	21	eximi 1547	. . . 4 |- ( E. x ( x = y /\ ps ) -> E. x ps )
23	20, 22	syl6bir 163	. . 3 |- ( ph -> ( ch -> E. x ps ) )
24	4, 18, 23	exlimd2 1542	. 2 |- ( ph -> ( E. y ch -> E. x ps ) )
25	15, 24	impbid 128	1 |- ( ph -> ( E. x ps <-> E. y ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1297   = wceq 1299   E.wex 1436

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  cbvexd  1862