Theorem cbvreu 2834

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/yph
cbvral.2	|- F/xps
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 1619	. . . 4 |- F/z(x e. A /\ ph)
2	1	sb8eu 2222	. . 3 |- (E!x(x e. A /\ ph) <-> E!z[z / x](x e. A /\ ph))
3		sban 2069	. . . 4 |- ([z / x](x e. A /\ ph) <-> ([z / x]x e. A /\ [z / x]ph))
4	3	eubii 2213	. . 3 |- (E!z[z / x](x e. A /\ ph) <-> E!z([z / x]x e. A /\ [z / x]ph))
5		clelsb1 2455	. . . . . 6 |- ([z / x]x e. A <-> z e. A)
6	5	anbi1i 676	. . . . 5 |- (([z / x]x e. A /\ [z / x]ph) <-> (z e. A /\ [z / x]ph))
7	6	eubii 2213	. . . 4 |- (E!z([z / x]x e. A /\ [z / x]ph) <-> E!z(z e. A /\ [z / x]ph))
8		nfv 1619	. . . . . 6 |- F/y z e. A
9		cbvral.1	. . . . . . 7 |- F/yph
10	9	nfsb 2109	. . . . . 6 |- F/y[z / x]ph
11	8, 10	nfan 1824	. . . . 5 |- F/y(z e. A /\ [z / x]ph)
12		nfv 1619	. . . . 5 |- F/z(y e. A /\ ps)
13		eleq1 2413	. . . . . 6 |- (z = y -> (z e. A <-> y e. A))
14		sbequ 2060	. . . . . . 7 |- (z = y -> ([z / x]ph <-> [y / x]ph))
15		cbvral.2	. . . . . . . 8 |- F/xps
16		cbvral.3	. . . . . . . 8 |- (x = y -> (ph <-> ps))
17	15, 16	sbie 2038	. . . . . . 7 |- ([y / x]ph <-> ps)
18	14, 17	syl6bb 252	. . . . . 6 |- (z = y -> ([z / x]ph <-> ps))
19	13, 18	anbi12d 691	. . . . 5 |- (z = y -> ((z e. A /\ [z / x]ph) <-> (y e. A /\ ps)))
20	11, 12, 19	cbveu 2224	. . . 4 |- (E!z(z e. A /\ [z / x]ph) <-> E!y(y e. A /\ ps))
21	7, 20	bitri 240	. . 3 |- (E!z([z / x]x e. A /\ [z / x]ph) <-> E!y(y e. A /\ ps))
22	2, 4, 21	3bitri 262	. 2 |- (E!x(x e. A /\ ph) <-> E!y(y e. A /\ ps))
23		df-reu 2622	. 2 |- (E!x e. A ph <-> E!x(x e. A /\ ph))
24		df-reu 2622	. 2 |- (E!y e. A ps <-> E!y(y e. A /\ ps))
25	22, 23, 24	3bitr4i 268	1 |- (E!x e. A ph <-> E!y e. A ps)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   F/wnf 1544   = wceq 1642  [wsb 1648   e. wcel 1710  E!weu 2204  E!wreu 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-cleq 2346  df-clel 2349  df-reu 2622

This theorem is referenced by:  cbvrmo  2835  cbvreuv  2838