Theorem cbvriotaw 7390

Description: Change bound variable in a restricted description binder. Version of cbvriota 7394 with a disjoint variable condition, which does not require ax-13 2401. (Contributed by NM, 18-Mar-2013.) Avoid ax-13 2401. (Revised by GG, 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 2843	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2		sbequ12 2286	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
3	1, 2	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)))
4		nfv 1946	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝜑)
5		nfv 1946	. . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
6		nfs1v 2192	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7	5, 6	nfan 1931	. . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
8	3, 4, 7	cbviotaw 6504	. . 3 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑))
9		eleq1w 2843	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10		cbvriotaw.2	. . . . . 6 ⊢ Ⅎ𝑥𝜓
11		cbvriotaw.3	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
12	10, 11	sbhypf 3509	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
13	9, 12	anbi12d 644	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
14		nfv 1946	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
15		cbvriotaw.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
16	15	nfsbv 2360	. . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
17	14, 16	nfan 1931	. . . 4 ⊢ Ⅎ𝑦(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)
18		nfv 1946	. . . 4 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓)
19	13, 17, 18	cbviotaw 6504	. . 3 ⊢ (℩𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑥]𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
20	8, 19	eqtri 2783	. 2 ⊢ (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
21		df-riota 7381	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
22		df-riota 7381	. 2 ⊢ (℩𝑦 ∈ 𝐴 𝜓) = (℩𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
23	20, 21, 22	3eqtr4i 2793	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  Ⅎwnf 1815  [wsb 2098   ∈ wcel 2144  ℩cio 6495  ℩crio 7380

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-sn 4585  df-uni 4867  df-iota 6497  df-riota 7381

This theorem is used by:  disjinfi  46343