Theorem cbvreu 2690

Description: Change the bound variable of a restricted unique existential 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 1516	. . . 4 |- F/ z ( x e. A /\ ph )
2	1	sb8eu 2027	. . 3 |- ( E! x ( x e. A /\ ph ) <-> E! z [ z / x ] ( x e. A /\ ph ) )
3		sban 1943	. . . 4 |- ( [ z / x ] ( x e. A /\ ph ) <-> ( [ z / x ] x e. A /\ [ z / x ] ph ) )
4	3	eubii 2023	. . 3 |- ( E! z [ z / x ] ( x e. A /\ ph ) <-> E! z ( [ z / x ] x e. A /\ [ z / x ] ph ) )
5		clelsb1 2271	. . . . . 6 |- ( [ z / x ] x e. A <-> z e. A )
6	5	anbi1i 454	. . . . 5 |- ( ( [ z / x ] x e. A /\ [ z / x ] ph ) <-> ( z e. A /\ [ z / x ] ph ) )
7	6	eubii 2023	. . . 4 |- ( E! z ( [ z / x ] x e. A /\ [ z / x ] ph ) <-> E! z ( z e. A /\ [ z / x ] ph ) )
8		nfv 1516	. . . . . 6 |- F/ y z e. A
9		cbvral.1	. . . . . . 7 |- F/ y ph
10	9	nfsb 1934	. . . . . 6 |- F/ y [ z / x ] ph
11	8, 10	nfan 1553	. . . . 5 |- F/ y ( z e. A /\ [ z / x ] ph )
12		nfv 1516	. . . . 5 |- F/ z ( y e. A /\ ps )
13		eleq1 2229	. . . . . 6 |- ( z = y -> ( z e. A <-> y e. A ) )
14		sbequ 1828	. . . . . . 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 1779	. . . . . . 7 |- ( [ y / x ] ph <-> ps )
18	14, 17	bitrdi 195	. . . . . 6 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
19	13, 18	anbi12d 465	. . . . 5 |- ( z = y -> ( ( z e. A /\ [ z / x ] ph ) <-> ( y e. A /\ ps ) ) )
20	11, 12, 19	cbveu 2038	. . . 4 |- ( E! z ( z e. A /\ [ z / x ] ph ) <-> E! y ( y e. A /\ ps ) )
21	7, 20	bitri 183	. . 3 |- ( E! z ( [ z / x ] x e. A /\ [ z / x ] ph ) <-> E! y ( y e. A /\ ps ) )
22	2, 4, 21	3bitri 205	. 2 |- ( E! x ( x e. A /\ ph ) <-> E! y ( y e. A /\ ps ) )
23		df-reu 2451	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
24		df-reu 2451	. 2 |- ( E! y e. A ps <-> E! y ( y e. A /\ ps ) )
25	22, 23, 24	3bitr4i 211	1 |- ( E! x e. A ph <-> E! y e. A ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   F/wnf 1448   [wsb 1750   E!weu 2014   e. wcel 2136   E!wreu 2446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-cleq 2158  df-clel 2161  df-reu 2451

This theorem is referenced by:  cbvrmo  2691  cbvreuv  2694