Theorem sb8iota 4346

Description: Variable substitution in description binder. Compare sb8eu 2222. (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 1619	. . . . . 6 ⊢ Ⅎw(φ ↔ x = z)
2	1	sb8 2092	. . . . 5 ⊢ (∀x(φ ↔ x = z) ↔ ∀w[w / x](φ ↔ x = z))
3		sbbi 2071	. . . . . . 7 ⊢ ([w / x](φ ↔ x = z) ↔ ([w / x]φ ↔ [w / x]x = z))
4		sb8iota.1	. . . . . . . . 9 ⊢ Ⅎyφ
5	4	nfsb 2109	. . . . . . . 8 ⊢ Ⅎy[w / x]φ
6		equsb3 2102	. . . . . . . . 9 ⊢ ([w / x]x = z ↔ w = z)
7		nfv 1619	. . . . . . . . 9 ⊢ Ⅎy w = z
8	6, 7	nfxfr 1570	. . . . . . . 8 ⊢ Ⅎy[w / x]x = z
9	5, 8	nfbi 1834	. . . . . . 7 ⊢ Ⅎy([w / x]φ ↔ [w / x]x = z)
10	3, 9	nfxfr 1570	. . . . . 6 ⊢ Ⅎy[w / x](φ ↔ x = z)
11		nfv 1619	. . . . . 6 ⊢ Ⅎw[y / x](φ ↔ x = z)
12		sbequ 2060	. . . . . 6 ⊢ (w = y → ([w / x](φ ↔ x = z) ↔ [y / x](φ ↔ x = z)))
13	10, 11, 12	cbval 1984	. . . . 5 ⊢ (∀w[w / x](φ ↔ x = z) ↔ ∀y[y / x](φ ↔ x = z))
14		equsb3 2102	. . . . . . 7 ⊢ ([y / x]x = z ↔ y = z)
15	14	sblbis 2072	. . . . . 6 ⊢ ([y / x](φ ↔ x = z) ↔ ([y / x]φ ↔ y = z))
16	15	albii 1566	. . . . 5 ⊢ (∀y[y / x](φ ↔ x = z) ↔ ∀y([y / x]φ ↔ y = z))
17	2, 13, 16	3bitri 262	. . . 4 ⊢ (∀x(φ ↔ x = z) ↔ ∀y([y / x]φ ↔ y = z))
18	17	abbii 2465	. . 3 ⊢ {z ∣ ∀x(φ ↔ x = z)} = {z ∣ ∀y([y / x]φ ↔ y = z)}
19	18	unieqi 3901	. 2 ⊢ ∪{z ∣ ∀x(φ ↔ x = z)} = ∪{z ∣ ∀y([y / x]φ ↔ y = z)}
20		dfiota2 4340	. 2 ⊢ (℩xφ) = ∪{z ∣ ∀x(φ ↔ x = z)}
21		dfiota2 4340	. 2 ⊢ (℩y[y / x]φ) = ∪{z ∣ ∀y([y / x]φ ↔ y = z)}
22	19, 20, 21	3eqtr4i 2383	1 ⊢ (℩xφ) = (℩y[y / x]φ)

Syntax hints:   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  [wsb 1648  {cab 2339  ∪cuni 3891  ℩cio 4337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-sn 3741  df-uni 3892  df-iota 4339

This theorem is referenced by: (None)