Theorem cbvreu 3409

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker cbvreuw 3396 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbvrmo.1	⊢ Ⅎ𝑦𝜑
cbvrmo.2	⊢ Ⅎ𝑥𝜓
cbvrmo.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 1937	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑)
2	1	sb8eu 2630	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑧[𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
3		sban 2116	. . . 4 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
4	3	eubii 2615	. . 3 ⊢ (∃!𝑧[𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
5		clelsb1 2892	. . . . . 6 ⊢ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
6	5	anbi1i 635	. . . . 5 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
7	6	eubii 2615	. . . 4 ⊢ (∃!𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
8		nfv 1937	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
9		cbvrmo.1	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
10	9	nfsb 2557	. . . . . 6 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
11	8, 10	nfan 1922	. . . . 5 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
12		nfv 1937	. . . . 5 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓)
13		eleq1w 2848	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
14		sbequ 2119	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15		cbvrmo.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝜓
16		cbvrmo.3	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
17	15, 16	sbie 2536	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
18	14, 17	bitrdi 290	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
19	13, 18	anbi12d 643	. . . . 5 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
20	11, 12, 19	cbveu 2637	. . . 4 ⊢ (∃!𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
21	7, 20	bitri 278	. . 3 ⊢ (∃!𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
22	2, 4, 21	3bitri 300	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
23		df-reu 3371	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
24		df-reu 3371	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
25	22, 23, 24	3bitr4i 306	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  Ⅎwnf 1806  [wsb 2093   ∈ wcel 2145  ∃!weu 2598  ∃!wreu 3368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-10 2178  ax-11 2194  ax-12 2215  ax-13 2406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clel 2840  df-reu 3371

This theorem is referenced by:  cbvrmo  3410  cbvreuv  3412