Theorem cbvreu 2500

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	⊢ Ⅎyφ
cbvral.2	⊢ Ⅎxψ
cbvral.3	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvreu	⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ)

Distinct variable groups:   x,A   y,A

Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem cbvreu

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1394	. . . 4 ⊢ Ⅎz(x ∈ A ∧ φ)
2	1	sb8eu 1886	. . 3 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!z[z / x](x ∈ A ∧ φ))
3		sban 1802	. . . 4 ⊢ ([z / x](x ∈ A ∧ φ) ↔ ([z / x]x ∈ A ∧ [z / x]φ))
4	3	eubii 1882	. . 3 ⊢ (∃!z[z / x](x ∈ A ∧ φ) ↔ ∃!z([z / x]x ∈ A ∧ [z / x]φ))
5		clelsb3 2115	. . . . . 6 ⊢ ([z / x]x ∈ A ↔ z ∈ A)
6	5	anbi1i 431	. . . . 5 ⊢ (([z / x]x ∈ A ∧ [z / x]φ) ↔ (z ∈ A ∧ [z / x]φ))
7	6	eubii 1882	. . . 4 ⊢ (∃!z([z / x]x ∈ A ∧ [z / x]φ) ↔ ∃!z(z ∈ A ∧ [z / x]φ))
8		nfv 1394	. . . . . 6 ⊢ Ⅎy z ∈ A
9		cbvral.1	. . . . . . 7 ⊢ Ⅎyφ
10	9	nfsb 1795	. . . . . 6 ⊢ Ⅎy[z / x]φ
11	8, 10	nfan 1430	. . . . 5 ⊢ Ⅎy(z ∈ A ∧ [z / x]φ)
12		nfv 1394	. . . . 5 ⊢ Ⅎz(y ∈ A ∧ ψ)
13		eleq1 2073	. . . . . 6 ⊢ (z = y → (z ∈ A ↔ y ∈ A))
14		sbequ 1694	. . . . . . 7 ⊢ (z = y → ([z / x]φ ↔ [y / x]φ))
15		cbvral.2	. . . . . . . 8 ⊢ Ⅎxψ
16		cbvral.3	. . . . . . . 8 ⊢ (x = y → (φ ↔ ψ))
17	15, 16	sbie 1647	. . . . . . 7 ⊢ ([y / x]φ ↔ ψ)
18	14, 17	syl6bb 185	. . . . . 6 ⊢ (z = y → ([z / x]φ ↔ ψ))
19	13, 18	anbi12d 442	. . . . 5 ⊢ (z = y → ((z ∈ A ∧ [z / x]φ) ↔ (y ∈ A ∧ ψ)))
20	11, 12, 19	cbveu 1897	. . . 4 ⊢ (∃!z(z ∈ A ∧ [z / x]φ) ↔ ∃!y(y ∈ A ∧ ψ))
21	7, 20	bitri 173	. . 3 ⊢ (∃!z([z / x]x ∈ A ∧ [z / x]φ) ↔ ∃!y(y ∈ A ∧ ψ))
22	2, 4, 21	3bitri 195	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!y(y ∈ A ∧ ψ))
23		df-reu 2282	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
24		df-reu 2282	. 2 ⊢ (∃!y ∈ A ψ ↔ ∃!y(y ∈ A ∧ ψ))
25	22, 23, 24	3bitr4i 201	1 ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  Ⅎwnf 1322   ∈ wcel 1366  [wsb 1618  ∃!weu 1873  ∃!wreu 2277

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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995

This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-cleq 2006  df-clel 2009  df-reu 2282

This theorem is referenced by:  cbvrmo  2501  cbvreuv  2504