Theorem cbvreuw 3443

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreu 3447 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvreuw.1	⊢ Ⅎ𝑦𝜑
cbvreuw.2	⊢ Ⅎ𝑥𝜓
cbvreuw.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable group:   𝑥,𝐴,𝑦

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

Proof of Theorem cbvreuw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1915	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑)
2	1	sb8euv 2685	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑧[𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
3		sban 2086	. . . 4 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
4	3	eubii 2670	. . 3 ⊢ (∃!𝑧[𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
5		clelsb3 2940	. . . . . 6 ⊢ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
6	5	anbi1i 625	. . . . 5 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
7	6	eubii 2670	. . . 4 ⊢ (∃!𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
8		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
9		cbvreuw.1	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
10	9	nfsbv 2349	. . . . . 6 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
11	8, 10	nfan 1900	. . . . 5 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
12		nfv 1915	. . . . 5 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓)
13		eleq1w 2895	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
14		sbequ 2090	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15		cbvreuw.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝜓
16		cbvreuw.3	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
17	15, 16	sbiev 2330	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
18	14, 17	syl6bb 289	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
19	13, 18	anbi12d 632	. . . . 5 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
20	11, 12, 19	cbveuw 2690	. . . 4 ⊢ (∃!𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
21	7, 20	bitri 277	. . 3 ⊢ (∃!𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
22	2, 4, 21	3bitri 299	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
23		df-reu 3145	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
24		df-reu 3145	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
25	22, 23, 24	3bitr4i 305	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  Ⅎwnf 1784  [wsb 2069   ∈ wcel 2114  ∃!weu 2653  ∃!wreu 3140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clel 2893  df-reu 3145

This theorem is referenced by:  cbvrmow  3444  cbvreuvw  3451  reu8nf  3860  poimirlem25  34932