Theorem cbviota 5201

Description: Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Hypotheses

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

Assertion

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

Proof of Theorem cbviota

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

Step	Hyp	Ref	Expression
1		nfv 1539	. . . . . 6 ⊢ Ⅎ𝑧(𝜑 ↔ 𝑥 = 𝑤)
2		nfs1v 1951	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		nfv 1539	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝑤
4	2, 3	nfbi 1600	. . . . . 6 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)
5		sbequ12 1782	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
6		equequ1 1723	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑤 ↔ 𝑧 = 𝑤))
7	5, 6	bibi12d 235	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑤) ↔ ([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)))
8	1, 4, 7	cbval 1765	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤))
9		cbviota.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
10	9	nfsb 1958	. . . . . . 7 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
11		nfv 1539	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 = 𝑤
12	10, 11	nfbi 1600	. . . . . 6 ⊢ Ⅎ𝑦([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)
13		nfv 1539	. . . . . 6 ⊢ Ⅎ𝑧(𝜓 ↔ 𝑦 = 𝑤)
14		sbequ 1851	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15		cbviota.3	. . . . . . . . 9 ⊢ Ⅎ𝑥𝜓
16		cbviota.1	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
17	15, 16	sbie 1802	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
18	14, 17	bitrdi 196	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
19		equequ1 1723	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑦 = 𝑤))
20	18, 19	bibi12d 235	. . . . . 6 ⊢ (𝑧 = 𝑦 → (([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤) ↔ (𝜓 ↔ 𝑦 = 𝑤)))
21	12, 13, 20	cbval 1765	. . . . 5 ⊢ (∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤) ↔ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤))
22	8, 21	bitri 184	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤))
23	22	abbii 2305	. . 3 ⊢ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
24	23	unieqi 3834	. 2 ⊢ ∪ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)} = ∪ {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
25		dfiota2 5197	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)}
26		dfiota2 5197	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
27	24, 25, 26	3eqtr4i 2220	1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364  Ⅎwnf 1471  [wsb 1773  {cab 2175  ∪ cuni 3824  ℩cio 5194

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-sn 3613  df-uni 3825  df-iota 5196

This theorem is referenced by:  cbviotav  5202  cbvriota  5862