Theorem sb8iota 5160

Description: Variable substitution in description binder. Compare sb8eu 2027. (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 1516	. . . . . 6 ⊢ Ⅎ𝑤(𝜑 ↔ 𝑥 = 𝑧)
2	1	sb8 1844	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
3		sbbi 1947	. . . . . . 7 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4		sb8iota.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
5	4	nfsb 1934	. . . . . . . 8 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
6		equsb3 1939	. . . . . . . . 9 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
7		nfv 1516	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 = 𝑧
8	6, 7	nfxfr 1462	. . . . . . . 8 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑥 = 𝑧
9	5, 8	nfbi 1577	. . . . . . 7 ⊢ Ⅎ𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
10	3, 9	nfxfr 1462	. . . . . 6 ⊢ Ⅎ𝑦[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
11		nfv 1516	. . . . . 6 ⊢ Ⅎ𝑤[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
12		sbequ 1828	. . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)))
13	10, 11, 12	cbval 1742	. . . . 5 ⊢ (∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
14		equsb3 1939	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
15	14	sblbis 1948	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
16	15	albii 1458	. . . . 5 ⊢ (∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
17	2, 13, 16	3bitri 205	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
18	17	abbii 2282	. . 3 ⊢ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
19	18	unieqi 3799	. 2 ⊢ ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∪ {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
20		dfiota2 5154	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
21		dfiota2 5154	. 2 ⊢ (℩𝑦[𝑦 / 𝑥]𝜑) = ∪ {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
22	19, 20, 21	3eqtr4i 2196	1 ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 104  ∀wal 1341   = wceq 1343  Ⅎwnf 1448  [wsb 1750  {cab 2151  ∪ cuni 3789  ℩cio 5151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-sn 3582  df-uni 3790  df-iota 5153

This theorem is referenced by: (None)