Theorem cbvreucsf 3200

Description: A more general version of cbvreuv 2837 that has no distinct variable rextrictions. Changes bound variables using implicit substitution. (Contributed by Andrew Salmon, 13-Jul-2011.)

Hypotheses

Ref	Expression
cbvralcsf.1	⊢ ℲyA
cbvralcsf.2	⊢ ℲxB
cbvralcsf.3	⊢ Ⅎyφ
cbvralcsf.4	⊢ Ⅎxψ
cbvralcsf.5	⊢ (x = y → A = B)
cbvralcsf.6	⊢ (x = y → (φ ↔ ψ))

Assertion

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

Proof of Theorem cbvreucsf

Dummy variables v z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . 4 ⊢ Ⅎz(x ∈ A ∧ φ)
2		nfcsb1v 3168	. . . . . 6 ⊢ Ⅎx[z / x]A
3	2	nfcri 2483	. . . . 5 ⊢ Ⅎx z ∈ [z / x]A
4		nfs1v 2106	. . . . 5 ⊢ Ⅎx[z / x]φ
5	3, 4	nfan 1824	. . . 4 ⊢ Ⅎx(z ∈ [z / x]A ∧ [z / x]φ)
6		id 19	. . . . . 6 ⊢ (x = z → x = z)
7		csbeq1a 3144	. . . . . 6 ⊢ (x = z → A = [z / x]A)
8	6, 7	eleq12d 2421	. . . . 5 ⊢ (x = z → (x ∈ A ↔ z ∈ [z / x]A))
9		sbequ12 1919	. . . . 5 ⊢ (x = z → (φ ↔ [z / x]φ))
10	8, 9	anbi12d 691	. . . 4 ⊢ (x = z → ((x ∈ A ∧ φ) ↔ (z ∈ [z / x]A ∧ [z / x]φ)))
11	1, 5, 10	cbveu 2224	. . 3 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!z(z ∈ [z / x]A ∧ [z / x]φ))
12		nfcv 2489	. . . . . . 7 ⊢ Ⅎyz
13		cbvralcsf.1	. . . . . . 7 ⊢ ℲyA
14	12, 13	nfcsb 3170	. . . . . 6 ⊢ Ⅎy[z / x]A
15	14	nfcri 2483	. . . . 5 ⊢ Ⅎy z ∈ [z / x]A
16		cbvralcsf.3	. . . . . 6 ⊢ Ⅎyφ
17	16	nfsb 2109	. . . . 5 ⊢ Ⅎy[z / x]φ
18	15, 17	nfan 1824	. . . 4 ⊢ Ⅎy(z ∈ [z / x]A ∧ [z / x]φ)
19		nfv 1619	. . . 4 ⊢ Ⅎz(y ∈ B ∧ ψ)
20		id 19	. . . . . 6 ⊢ (z = y → z = y)
21		csbeq1 3139	. . . . . . 7 ⊢ (z = y → [z / x]A = [y / x]A)
22		sbsbc 3050	. . . . . . . . 9 ⊢ ([y / x]v ∈ A ↔ [̣y / x]̣v ∈ A)
23	22	abbii 2465	. . . . . . . 8 ⊢ {v ∣ [y / x]v ∈ A} = {v ∣ [̣y / x]̣v ∈ A}
24		cbvralcsf.2	. . . . . . . . . . . 12 ⊢ ℲxB
25	24	nfcri 2483	. . . . . . . . . . 11 ⊢ Ⅎx v ∈ B
26		cbvralcsf.5	. . . . . . . . . . . 12 ⊢ (x = y → A = B)
27	26	eleq2d 2420	. . . . . . . . . . 11 ⊢ (x = y → (v ∈ A ↔ v ∈ B))
28	25, 27	sbie 2038	. . . . . . . . . 10 ⊢ ([y / x]v ∈ A ↔ v ∈ B)
29	28	bicomi 193	. . . . . . . . 9 ⊢ (v ∈ B ↔ [y / x]v ∈ A)
30	29	abbi2i 2464	. . . . . . . 8 ⊢ B = {v ∣ [y / x]v ∈ A}
31		df-csb 3137	. . . . . . . 8 ⊢ [y / x]A = {v ∣ [̣y / x]̣v ∈ A}
32	23, 30, 31	3eqtr4ri 2384	. . . . . . 7 ⊢ [y / x]A = B
33	21, 32	syl6eq 2401	. . . . . 6 ⊢ (z = y → [z / x]A = B)
34	20, 33	eleq12d 2421	. . . . 5 ⊢ (z = y → (z ∈ [z / x]A ↔ y ∈ B))
35		sbequ 2060	. . . . . 6 ⊢ (z = y → ([z / x]φ ↔ [y / x]φ))
36		cbvralcsf.4	. . . . . . 7 ⊢ Ⅎxψ
37		cbvralcsf.6	. . . . . . 7 ⊢ (x = y → (φ ↔ ψ))
38	36, 37	sbie 2038	. . . . . 6 ⊢ ([y / x]φ ↔ ψ)
39	35, 38	syl6bb 252	. . . . 5 ⊢ (z = y → ([z / x]φ ↔ ψ))
40	34, 39	anbi12d 691	. . . 4 ⊢ (z = y → ((z ∈ [z / x]A ∧ [z / x]φ) ↔ (y ∈ B ∧ ψ)))
41	18, 19, 40	cbveu 2224	. . 3 ⊢ (∃!z(z ∈ [z / x]A ∧ [z / x]φ) ↔ ∃!y(y ∈ B ∧ ψ))
42	11, 41	bitri 240	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ ∃!y(y ∈ B ∧ ψ))
43		df-reu 2621	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
44		df-reu 2621	. 2 ⊢ (∃!y ∈ B ψ ↔ ∃!y(y ∈ B ∧ ψ))
45	42, 43, 44	3bitr4i 268	1 ⊢ (∃!x ∈ A φ ↔ ∃!y ∈ B ψ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃!weu 2204  {cab 2339  Ⅎwnfc 2476  ∃!wreu 2616  [̣wsbc 3046  [csb 3136

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-reu 2621  df-sbc 3047  df-csb 3137

This theorem is referenced by: (None)