Theorem cbvriotaw 7221

Description: Change bound variable in a restricted description binder. Version of cbvriota 7226 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 18-Mar-2013.) (Revised by Gino Giotto, 26-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
cbvriotaw	⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)

Distinct variable group:   𝑥,𝐴,𝑦

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

Proof of Theorem cbvriotaw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2821	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2		sbequ12 2247	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
3	1, 2	anbi12d 630	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)))
4		nfv 1918	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑)
5		nfv 1918	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
6		nfs1v 2155	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7	5, 6	nfan 1903	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
8	3, 4, 7	cbviotaw 6383	. . 3 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
9		eleq1w 2821	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10		cbvriotaw.2	. . . . . 6 ⊢ Ⅎ𝑥𝜓
11		cbvriotaw.3	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
12	10, 11	sbhypf 3481	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
13	9, 12	anbi12d 630	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
14		nfv 1918	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
15		cbvriotaw.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
16	15	nfsbv 2328	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
17	14, 16	nfan 1903	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
18		nfv 1918	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓)
19	13, 17, 18	cbviotaw 6383	. . 3 ⊢ (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
20	8, 19	eqtri 2766	. 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
21		df-riota 7212	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
22		df-riota 7212	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
23	20, 21, 22	3eqtr4i 2776	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  Ⅎwnf 1787  [wsb 2068   ∈ wcel 2108  ℩cio 6374  ℩crio 7211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-sn 4559  df-uni 4837  df-iota 6376  df-riota 7212

This theorem is referenced by:  cbvriotavwOLD  7223  disjinfi  42620