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	⊢ 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 1619	. . . 4 ⊢ Ⅎz(x ∈ A ∧ φ)
2	1	sb8eu 2222	. . 3 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!z[z / x](x ∈ A ∧ φ))
3		sban 2069	. . . 4 ⊢ ([z / x](x ∈ A ∧ φ) ↔ ([z / x]x ∈ A ∧ [z / x]φ))
4	3	eubii 2213	. . 3 ⊢ (∃!z[z / x](x ∈ A ∧ φ) ↔ ∃!z([z / x]x ∈ A ∧ [z / x]φ))
5		clelsb1 2455	. . . . . 6 ⊢ ([z / x]x ∈ A ↔ z ∈ A)
6	5	anbi1i 676	. . . . 5 ⊢ (([z / x]x ∈ A ∧ [z / x]φ) ↔ (z ∈ A ∧ [z / x]φ))
7	6	eubii 2213	. . . 4 ⊢ (∃!z([z / x]x ∈ A ∧ [z / x]φ) ↔ ∃!z(z ∈ A ∧ [z / x]φ))
8		nfv 1619	. . . . . 6 ⊢ Ⅎy z ∈ A
9		cbvral.1	. . . . . . 7 ⊢ Ⅎyφ
10	9	nfsb 2109	. . . . . 6 ⊢ Ⅎy[z / x]φ
11	8, 10	nfan 1824	. . . . 5 ⊢ Ⅎy(z ∈ A ∧ [z / x]φ)
12		nfv 1619	. . . . 5 ⊢ Ⅎz(y ∈ A ∧ ψ)
13		eleq1 2413	. . . . . 6 ⊢ (z = y → (z ∈ A ↔ y ∈ A))
14		sbequ 2060	. . . . . . 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 2038	. . . . . . 7 ⊢ ([y / x]φ ↔ ψ)
18	14, 17	syl6bb 252	. . . . . 6 ⊢ (z = y → ([z / x]φ ↔ ψ))
19	13, 18	anbi12d 691	. . . . 5 ⊢ (z = y → ((z ∈ A ∧ [z / x]φ) ↔ (y ∈ A ∧ ψ)))
20	11, 12, 19	cbveu 2224	. . . 4 ⊢ (∃!z(z ∈ A ∧ [z / x]φ) ↔ ∃!y(y ∈ A ∧ ψ))
21	7, 20	bitri 240	. . 3 ⊢ (∃!z([z / x]x ∈ A ∧ [z / x]φ) ↔ ∃!y(y ∈ A ∧ ψ))
22	2, 4, 21	3bitri 262	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!y(y ∈ A ∧ ψ))
23		df-reu 2622	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
24		df-reu 2622	. 2 ⊢ (∃!y ∈ A ψ ↔ ∃!y(y ∈ A ∧ ψ))
25	22, 23, 24	3bitr4i 268	1 ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ A ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃!weu 2204  ∃!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