Theorem sb8iota 5095

Description: Variable substitution in description binder. Compare sb8eu 2012. (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 1508	. . . . . 6 ⊢ Ⅎ𝑤(𝜑 ↔ 𝑥 = 𝑧)
2	1	sb8 1828	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
3		sbbi 1932	. . . . . . 7 ⊢ ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧))
4		sb8iota.1	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜑
5	4	nfsb 1919	. . . . . . . 8 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝜑
6		equsb3 1924	. . . . . . . . 9 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
7		nfv 1508	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑤 = 𝑧
8	6, 7	nfxfr 1450	. . . . . . . 8 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑥 = 𝑧
9	5, 8	nfbi 1568	. . . . . . 7 ⊢ Ⅎ𝑦([𝑤 / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝑥 = 𝑧)
10	3, 9	nfxfr 1450	. . . . . 6 ⊢ Ⅎ𝑦[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
11		nfv 1508	. . . . . 6 ⊢ Ⅎ𝑤[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)
12		sbequ 1812	. . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ [𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧)))
13	10, 11, 12	cbval 1727	. . . . 5 ⊢ (∀𝑤[𝑤 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑦[𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧))
14		equsb3 1924	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)
15	14	sblbis 1933	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ↔ 𝑥 = 𝑧) ↔ ([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧))
16	15	albii 1446	. . . . 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 2255	. . 3 ⊢ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
19	18	unieqi 3746	. 2 ⊢ ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = ∪ {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
20		dfiota2 5089	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
21		dfiota2 5089	. 2 ⊢ (℩𝑦[𝑦 / 𝑥]𝜑) = ∪ {𝑧 ∣ ∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝑦 = 𝑧)}
22	19, 20, 21	3eqtr4i 2170	1 ⊢ (℩𝑥𝜑) = (℩𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 104  ∀wal 1329   = wceq 1331  Ⅎwnf 1436  [wsb 1735  {cab 2125  ∪ cuni 3736  ℩cio 5086

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-sn 3533  df-uni 3737  df-iota 5088

This theorem is referenced by: (None)