Theorem cbviota 4872

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 1421	. . . . . 6 ⊢ Ⅎ𝑧(𝜑 ↔ 𝑥 = 𝑤)
2		nfs1v 1815	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		nfv 1421	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝑤
4	2, 3	nfbi 1481	. . . . . 6 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)
5		sbequ12 1654	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
6		equequ1 1598	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑤 ↔ 𝑧 = 𝑤))
7	5, 6	bibi12d 224	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑤) ↔ ([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)))
8	1, 4, 7	cbval 1637	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤))
9		cbviota.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
10	9	nfsb 1822	. . . . . . 7 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
11		nfv 1421	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 = 𝑤
12	10, 11	nfbi 1481	. . . . . 6 ⊢ Ⅎ𝑦([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)
13		nfv 1421	. . . . . 6 ⊢ Ⅎ𝑧(𝜓 ↔ 𝑦 = 𝑤)
14		sbequ 1721	. . . . . . . 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 1674	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
18	14, 17	syl6bb 185	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
19		equequ1 1598	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑦 = 𝑤))
20	18, 19	bibi12d 224	. . . . . 6 ⊢ (𝑧 = 𝑦 → (([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤) ↔ (𝜓 ↔ 𝑦 = 𝑤)))
21	12, 13, 20	cbval 1637	. . . . 5 ⊢ (∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤) ↔ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤))
22	8, 21	bitri 173	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤))
23	22	abbii 2153	. . 3 ⊢ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
24	23	unieqi 3590	. 2 ⊢ ∪ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)} = ∪ {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
25		dfiota2 4868	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)}
26		dfiota2 4868	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
27	24, 25, 26	3eqtr4i 2070	1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241   = wceq 1243  Ⅎwnf 1349  [wsb 1645  {cab 2026  ∪ cuni 3580  ℩cio 4865

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-sn 3381  df-uni 3581  df-iota 4867

This theorem is referenced by:  cbviotav  4873  cbvriota  5478