Theorem cbvreu 3169

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 1843	. . . 4 |- F/ z ( x e. A /\ ph )
2	1	sb8eu 2503	. . 3 |- ( E! x ( x e. A /\ ph ) <-> E! z [ z / x ] ( x e. A /\ ph ) )
3		sban 2399	. . . 4 |- ( [ z / x ] ( x e. A /\ ph ) <-> ( [ z / x ] x e. A /\ [ z / x ] ph ) )
4	3	eubii 2492	. . 3 |- ( E! z [ z / x ] ( x e. A /\ ph ) <-> E! z ( [ z / x ] x e. A /\ [ z / x ] ph ) )
5		clelsb3 2729	. . . . . 6 |- ( [ z / x ] x e. A <-> z e. A )
6	5	anbi1i 731	. . . . 5 |- ( ( [ z / x ] x e. A /\ [ z / x ] ph ) <-> ( z e. A /\ [ z / x ] ph ) )
7	6	eubii 2492	. . . 4 |- ( E! z ( [ z / x ] x e. A /\ [ z / x ] ph ) <-> E! z ( z e. A /\ [ z / x ] ph ) )
8		nfv 1843	. . . . . 6 |- F/ y z e. A
9		cbvral.1	. . . . . . 7 |- F/ y ph
10	9	nfsb 2440	. . . . . 6 |- F/ y [ z / x ] ph
11	8, 10	nfan 1828	. . . . 5 |- F/ y ( z e. A /\ [ z / x ] ph )
12		nfv 1843	. . . . 5 |- F/ z ( y e. A /\ ps )
13		eleq1 2689	. . . . . 6 |- ( z = y -> ( z e. A <-> y e. A ) )
14		sbequ 2376	. . . . . . 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 2408	. . . . . . 7 |- ( [ y / x ] ph <-> ps )
18	14, 17	syl6bb 276	. . . . . 6 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
19	13, 18	anbi12d 747	. . . . 5 |- ( z = y -> ( ( z e. A /\ [ z / x ] ph ) <-> ( y e. A /\ ps ) ) )
20	11, 12, 19	cbveu 2505	. . . 4 |- ( E! z ( z e. A /\ [ z / x ] ph ) <-> E! y ( y e. A /\ ps ) )
21	7, 20	bitri 264	. . 3 |- ( E! z ( [ z / x ] x e. A /\ [ z / x ] ph ) <-> E! y ( y e. A /\ ps ) )
22	2, 4, 21	3bitri 286	. 2 |- ( E! x ( x e. A /\ ph ) <-> E! y ( y e. A /\ ps ) )
23		df-reu 2919	. 2 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
24		df-reu 2919	. 2 |- ( E! y e. A ps <-> E! y ( y e. A /\ ps ) )
25	22, 23, 24	3bitr4i 292	1 |- ( E! x e. A ph <-> E! y e. A ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   F/wnf 1708   [wsb 1880   e. wcel 1990   E!weu 2470   E!wreu 2914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-cleq 2615  df-clel 2618  df-reu 2919

This theorem is referenced by:  cbvrmo  3170  cbvreuv  3173  reu8nf  3516  poimirlem25  33434