Theorem cbviota 6465

Description: Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker cbviotaw 6463 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.)

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 1916	. . . . . 6 ⊢ Ⅎ𝑧(𝜑 ↔ 𝑥 = 𝑤)
2		nfs1v 2162	. . . . . . 7 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝑤
4	2, 3	nfbi 1905	. . . . . 6 ⊢ Ⅎ𝑥([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)
5		sbequ12 2259	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
6		equequ1 2027	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝑤 ↔ 𝑧 = 𝑤))
7	5, 6	bibi12d 345	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝜑 ↔ 𝑥 = 𝑤) ↔ ([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)))
8	1, 4, 7	cbvalv1 2346	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤))
9		cbviota.2	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
10	9	nfsb 2528	. . . . . . 7 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
11		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑦 𝑧 = 𝑤
12	10, 11	nfbi 1905	. . . . . 6 ⊢ Ⅎ𝑦([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤)
13		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑧(𝜓 ↔ 𝑦 = 𝑤)
14		sbequ 2089	. . . . . . . 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 2507	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
18	14, 17	bitrdi 287	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ 𝜓))
19		equequ1 2027	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑦 = 𝑤))
20	18, 19	bibi12d 345	. . . . . 6 ⊢ (𝑧 = 𝑦 → (([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤) ↔ (𝜓 ↔ 𝑦 = 𝑤)))
21	12, 13, 20	cbvalv1 2346	. . . . 5 ⊢ (∀𝑧([𝑧 / 𝑥]𝜑 ↔ 𝑧 = 𝑤) ↔ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤))
22	8, 21	bitri 275	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑤) ↔ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤))
23	22	abbii 2804	. . 3 ⊢ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
24	23	unieqi 4877	. 2 ⊢ ∪ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)} = ∪ {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
25		dfiota2 6457	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑤 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑤)}
26		dfiota2 6457	. 2 ⊢ (℩𝑦𝜓) = ∪ {𝑤 ∣ ∀𝑦(𝜓 ↔ 𝑦 = 𝑤)}
27	24, 25, 26	3eqtr4i 2770	1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542  Ⅎwnf 1785  [wsb 2068  {cab 2715  ∪ cuni 4865  ℩cio 6454

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-13 2377  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-ss 3920  df-sn 4583  df-uni 4866  df-iota 6456

This theorem is referenced by:  cbviotav  6466  cbvriota  7338