Theorem cbvriota 5933

Description: Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
cbvriota	⊢ (℩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 cbvriota

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2270	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2		sbequ12 1795	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
3	1, 2	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)))
4		nfv 1552	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑)
5		nfv 1552	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
6		nfs1v 1968	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7	5, 6	nfan 1589	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
8	3, 4, 7	cbviota 5256	. . 3 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
9		eleq1 2270	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10		sbequ 1864	. . . . . 6 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
11		cbvriota.2	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
12		cbvriota.3	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
13	11, 12	sbie 1815	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
14	10, 13	bitrdi 196	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
15	9, 14	anbi12d 473	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
16		nfv 1552	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
17		cbvriota.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
18	17	nfsb 1975	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
19	16, 18	nfan 1589	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
20		nfv 1552	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓)
21	15, 19, 20	cbviota 5256	. . 3 ⊢ (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
22	8, 21	eqtri 2228	. 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
23		df-riota 5922	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
24		df-riota 5922	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
25	22, 23, 24	3eqtr4i 2238	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  Ⅎwnf 1484  [wsb 1786   ∈ wcel 2178  ℩cio 5249  ℩crio 5921

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-sn 3649  df-uni 3865  df-iota 5251  df-riota 5922

This theorem is referenced by:  cbvriotav  5934