Theorem sb8iota 6403

Description: Variable substitution in description binder. Compare sb8eu 2600. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 18-Mar-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
sb8iota.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb8iota	⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb8iota

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

Step	Hyp	Ref	Expression
1		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑤(𝜑 ↔ 𝑥 = 𝑧)
2	1	sb8 2521	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
3		sbbi 2305	. . . . . . 7 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4		sb8iota.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
5	4	nfsb 2527	. . . . . . . 8 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
6		equsb3 2101	. . . . . . . . 9 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
7		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 = 𝑧
8	6, 7	nfxfr 1855	. . . . . . . 8 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑥 = 𝑧
9	5, 8	nfbi 1906	. . . . . . 7 ⊢ Ⅎ𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
10	3, 9	nfxfr 1855	. . . . . 6 ⊢ Ⅎ𝑦[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
11		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑤[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
12		sbequ 2086	. . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)))
13	10, 11, 12	cbvalv1 2338	. . . . 5 ⊢ (∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
14		equsb3 2101	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
15	14	sblbis 2306	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
16	15	albii 1822	. . . . 5 ⊢ (∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
17	2, 13, 16	3bitri 297	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
18	17	abbii 2808	. . 3 ⊢ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
19	18	unieqi 4852	. 2 ⊢ ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∪ {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
20		dfiota2 6392	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
21		dfiota2 6392	. 2 ⊢ (℩𝑦[𝑦 / 𝑥]𝜑) = ∪ {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
22	19, 20, 21	3eqtr4i 2776	1 ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  ∀wal 1537   = wceq 1539  Ⅎwnf 1786  [wsb 2067  {cab 2715  ∪ cuni 4839  ℩cio 6389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-sn 4562  df-uni 4840  df-iota 6391

This theorem is referenced by: (None)