Theorem sb8iota 4874

Description: Variable substitution in description binder. Compare sb8eu 1913. (Contributed by NM, 18-Mar-2013.)

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 1421	. . . . . 6 ⊢ Ⅎ𝑤(𝜑 ↔ 𝑥 = 𝑧)
2	1	sb8 1736	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
3		sbbi 1833	. . . . . . 7 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4		sb8iota.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
5	4	nfsb 1822	. . . . . . . 8 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
6		equsb3 1825	. . . . . . . . 9 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
7		nfv 1421	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 = 𝑧
8	6, 7	nfxfr 1363	. . . . . . . 8 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑥 = 𝑧
9	5, 8	nfbi 1481	. . . . . . 7 ⊢ Ⅎ𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
10	3, 9	nfxfr 1363	. . . . . 6 ⊢ Ⅎ𝑦[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
11		nfv 1421	. . . . . 6 ⊢ Ⅎ𝑤[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
12		sbequ 1721	. . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)))
13	10, 11, 12	cbval 1637	. . . . 5 ⊢ (∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
14		equsb3 1825	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
15	14	sblbis 1834	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
16	15	albii 1359	. . . . 5 ⊢ (∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
17	2, 13, 16	3bitri 195	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
18	17	abbii 2153	. . 3 ⊢ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
19	18	unieqi 3590	. 2 ⊢ ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∪ {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
20		dfiota2 4868	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
21		dfiota2 4868	. 2 ⊢ (℩𝑦[𝑦 / 𝑥]𝜑) = ∪ {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
22	19, 20, 21	3eqtr4i 2070	1 ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ 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: (None)