Theorem cbvreu 2727

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 1542	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑)
2	1	sb8eu 2058	. . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑧[𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑))
3		sban 1974	. . . 4 ⊢ ([𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
4	3	eubii 2054	. . 3 ⊢ (∃!𝑧[𝑧 / 𝑥](𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
5		clelsb1 2301	. . . . . 6 ⊢ ([𝑧 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
6	5	anbi1i 458	. . . . 5 ⊢ (([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
7	6	eubii 2054	. . . 4 ⊢ (∃!𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
8		nfv 1542	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
9		cbvral.1	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
10	9	nfsb 1965	. . . . . 6 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
11	8, 10	nfan 1579	. . . . 5 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
12		nfv 1542	. . . . 5 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓)
13		eleq1 2259	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
14		sbequ 1854	. . . . . . 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 1805	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
18	14, 17	bitrdi 196	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
19	13, 18	anbi12d 473	. . . . 5 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
20	11, 12, 19	cbveu 2069	. . . 4 ⊢ (∃!𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
21	7, 20	bitri 184	. . 3 ⊢ (∃!𝑧([𝑧 / 𝑥]𝑥 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
22	2, 4, 21	3bitri 206	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
23		df-reu 2482	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
24		df-reu 2482	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
25	22, 23, 24	3bitr4i 212	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  Ⅎwnf 1474  [wsb 1776  ∃!weu 2045   ∈ wcel 2167  ∃!wreu 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-cleq 2189  df-clel 2192  df-reu 2482

This theorem is referenced by:  cbvrmo  2728  cbvreuv  2731