Theorem cbvreucsf 3816

Description: A more general version of cbvreuv 3379 that has no distinct variable restrictions. 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 1873	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑)
2		nfcsb1v 3798	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴
3	2	nfcri 2920	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴
4		nfs1v 2202	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
5	3, 4	nfan 1862	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑)
6		id 22	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
7		csbeq1a 3789	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
8	6, 7	eleq12d 2854	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴))
9		sbequ12 2179	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
10	8, 9	anbi12d 621	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑)))
11	1, 5, 10	cbveu 2637	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑))
12		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑦𝑧
13		cbvralcsf.1	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
14	12, 13	nfcsb 3800	. . . . . 6 ⊢ Ⅎ𝑦⦋𝑧 / 𝑥⦌𝐴
15	14	nfcri 2920	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴
16		cbvralcsf.3	. . . . . 6 ⊢ Ⅎ𝑦𝜑
17	16	nfsb 2488	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
18	15, 17	nfan 1862	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑)
19		nfv 1873	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐵 ∧ 𝜓)
20		id 22	. . . . . 6 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
21		csbeq1 3783	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
22		sbsbc 3679	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑣 ∈ 𝐴)
23	22	abbii 2838	. . . . . . . 8 ⊢ {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴} = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴}
24		cbvralcsf.2	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐵
25	24	nfcri 2920	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑣 ∈ 𝐵
26		cbvralcsf.5	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
27	26	eleq2d 2845	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐵))
28	25, 27	sbie 2468	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥]𝑣 ∈ 𝐴 ↔ 𝑣 ∈ 𝐵)
29	28	bicomi 216	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝐵 ↔ [𝑦 / 𝑥]𝑣 ∈ 𝐴)
30	29	abbi2i 2899	. . . . . . . 8 ⊢ 𝐵 = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴}
31		df-csb 3781	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = {𝑣 ∣ [𝑦 / 𝑥]𝑣 ∈ 𝐴}
32	23, 30, 31	3eqtr4ri 2807	. . . . . . 7 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = 𝐵
33	21, 32	syl6eq 2824	. . . . . 6 ⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐴 = 𝐵)
34	20, 33	eleq12d 2854	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ↔ 𝑦 ∈ 𝐵))
35		sbequ 2035	. . . . . 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 2468	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
39	35, 38	syl6bb 279	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
40	34, 39	anbi12d 621	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜓)))
41	18, 19, 40	cbveu 2637	. . 3 ⊢ (∃!𝑧(𝑧 ∈ ⦋𝑧 / 𝑥⦌𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))
42	11, 41	bitri 267	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))
43		df-reu 3089	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
44		df-reu 3089	. 2 ⊢ (∃!𝑦 ∈ 𝐵 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐵 ∧ 𝜓))
45	42, 43, 44	3bitr4i 295	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507  Ⅎwnf 1746  [wsb 2015   ∈ wcel 2050  ∃!weu 2583  {cab 2752  Ⅎwnfc 2910  ∃!wreu 3084  [wsbc 3675  ⦋csb 3780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-reu 3089  df-sbc 3676  df-csb 3781

This theorem is referenced by: (None)