Theorem cbviotaw 6306

Description: Change bound variables in a description binder. Version of cbviota 6308 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
cbviotaw	⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbviotaw

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧(𝜑 ↔ 𝑥 = 𝑤)
2		nfs1v 2157	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝑤
4	2, 3	nfbi 1904	. . . . . 6 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)
5		sbequ12 2250	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
6		equequ1 2032	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑤 ↔ 𝑧 = 𝑤))
7	5, 6	bibi12d 349	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑤) ↔ ([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)))
8	1, 4, 7	cbvalv1 2350	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤))
9		cbviotaw.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
10	9	nfsbv 2338	. . . . . . 7 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
11		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 = 𝑤
12	10, 11	nfbi 1904	. . . . . 6 ⊢ Ⅎ𝑦([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)
13		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧(𝜓 ↔ 𝑦 = 𝑤)
14		cbviotaw.3	. . . . . . . 8 ⊢ Ⅎ𝑥𝜓
15		cbviotaw.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
16	14, 15	sbhypf 3471	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
17		equequ1 2032	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑦 = 𝑤))
18	16, 17	bibi12d 349	. . . . . 6 ⊢ (𝑧 = 𝑦 → (([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤) ↔ (𝜓 ↔ 𝑦 = 𝑤)))
19	12, 13, 18	cbvalv1 2350	. . . . 5 ⊢ (∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤) ↔ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤))
20	8, 19	bitri 278	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤))
21	20	abbii 2823	. . 3 ⊢ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
22	21	unieqi 4814	. 2 ⊢ ∪ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)} = ∪ {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
23		dfiota2 6300	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)}
24		dfiota2 6300	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
25	22, 23, 24	3eqtr4i 2791	1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538  Ⅎwnf 1785  [wsb 2069  {cab 2735  ∪ cuni 4801  ℩cio 6297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3867  df-ss 3877  df-sn 4526  df-uni 4802  df-iota 6299

This theorem is referenced by:  cbviotavw  6307  fvopab5  6796  cbvriotaw  7123