Theorem cbvreu 2531

Description: Change the bound variable of a restricted uniqueness quantifier using implicit substitution. (Contributed by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
cbvral.1	|- F/ y ph
cbvral.2	|- F/ x ps
cbvral.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvreu	|- ( E! x e. A ph <-> E! y e. A ps )

Distinct variable groups:   x, A   y, A

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

Proof of Theorem cbvreu

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1421	. . . 4 |- F/ z ( x e. A /\ ph )
2	1	sb8eu 1913	. . 3 |- ( E! x ( x e. A /\ ph ) <-> E! z [ z / x ] ( x e. A /\ ph ) )
3		sban 1829	. . . 4 |- ( [ z / x ] ( x e. A /\ ph ) <-> ( [ z / x ] x e. A /\ [ z / x ] ph ) )
4	3	eubii 1909	. . 3 |- ( E! z [ z / x ] ( x e. A /\ ph ) <-> E! z ( [ z / x ] x e. A /\ [ z / x ] ph ) )
5		clelsb3 2142	. . . . . 6 |- ( [ z / x ] x e. A <-> z e. A )
6	5	anbi1i 431	. . . . 5 |- ( ( [ z / x ] x e. A /\ [ z / x ] ph ) <-> ( z e. A /\ [ z / x ] ph ) )
7	6	eubii 1909	. . . 4 |- ( E! z ( [ z / x ] x e. A /\ [ z / x ] ph ) <-> E! z ( z e. A /\ [ z / x ] ph ) )
8		nfv 1421	. . . . . 6 |- F/ y z e. A
9		cbvral.1	. . . . . . 7 |- F/ y ph
10	9	nfsb 1822	. . . . . 6 |- F/ y [ z / x ] ph
11	8, 10	nfan 1457	. . . . 5 |- F/ y ( z e. A /\ [ z / x ] ph )
12		nfv 1421	. . . . 5 |- F/ z ( y e. A /\ ps )
13		eleq1 2100	. . . . . 6 |- ( z = y -> ( z e. A <-> y e. A ) )
14		sbequ 1721	. . . . . . 7 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
15		cbvral.2	. . . . . . . 8 |- F/ x ps
16		cbvral.3	. . . . . . . 8 |- ( x = y -> ( ph <-> ps ) )
17	15, 16	sbie 1674	. . . . . . 7 |- ( [ y / x ] ph <-> ps )
18	14, 17	syl6bb 185	. . . . . 6 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
19	13, 18	anbi12d 442	. . . . 5 |- ( z = y -> ( ( z e. A /\ [ z / x ] ph ) <-> ( y e. A /\ ps ) ) )
20	11, 12, 19	cbveu 1924	. . . 4 |- ( E! z ( z e. A /\ [ z / x ] ph ) <-> E! y ( y e. A /\ ps ) )
21	7, 20	bitri 173	. . 3 |- ( E! z ( [ z / x ] x e. A /\ [ z / x ] ph ) <-> E! y ( y e. A /\ ps ) )
22	2, 4, 21	3bitri 195	. 2 |- ( E! x ( x e. A /\ ph ) <-> E! y ( y e. A /\ ps ) )
23		df-reu 2313	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
24		df-reu 2313	. 2 |- ( E! y e. A ps <-> E! y ( y e. A /\ ps ) )
25	22, 23, 24	3bitr4i 201	1 |- ( E! x e. A ph <-> E! y e. A ps )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   F/wnf 1349   e. wcel 1393   [wsb 1645   E!weu 1900   E!wreu 2308

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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-cleq 2033  df-clel 2036  df-reu 2313

This theorem is referenced by:  cbvrmo  2532  cbvreuv  2535