Theorem cbviota 4345

Description: Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Hypotheses

Ref	Expression
cbviota.1	⊢ (x = y → (φ ↔ ψ))
cbviota.2	⊢ Ⅎyφ
cbviota.3	⊢ Ⅎxψ

Assertion

Ref	Expression
cbviota	⊢ (℩xφ) = (℩yψ)

Proof of Theorem cbviota

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

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . . 6 ⊢ Ⅎz(φ ↔ x = w)
2		nfs1v 2106	. . . . . . 7 ⊢ Ⅎx[z / x]φ
3		nfv 1619	. . . . . . 7 ⊢ Ⅎx z = w
4	2, 3	nfbi 1834	. . . . . 6 ⊢ Ⅎx([z / x]φ ↔ z = w)
5		sbequ12 1919	. . . . . . 7 ⊢ (x = z → (φ ↔ [z / x]φ))
6		equequ1 1684	. . . . . . 7 ⊢ (x = z → (x = w ↔ z = w))
7	5, 6	bibi12d 312	. . . . . 6 ⊢ (x = z → ((φ ↔ x = w) ↔ ([z / x]φ ↔ z = w)))
8	1, 4, 7	cbval 1984	. . . . 5 ⊢ (∀x(φ ↔ x = w) ↔ ∀z([z / x]φ ↔ z = w))
9		cbviota.2	. . . . . . . 8 ⊢ Ⅎyφ
10	9	nfsb 2109	. . . . . . 7 ⊢ Ⅎy[z / x]φ
11		nfv 1619	. . . . . . 7 ⊢ Ⅎy z = w
12	10, 11	nfbi 1834	. . . . . 6 ⊢ Ⅎy([z / x]φ ↔ z = w)
13		nfv 1619	. . . . . 6 ⊢ Ⅎz(ψ ↔ y = w)
14		sbequ 2060	. . . . . . . 8 ⊢ (z = y → ([z / x]φ ↔ [y / x]φ))
15		cbviota.3	. . . . . . . . 9 ⊢ Ⅎxψ
16		cbviota.1	. . . . . . . . 9 ⊢ (x = y → (φ ↔ ψ))
17	15, 16	sbie 2038	. . . . . . . 8 ⊢ ([y / x]φ ↔ ψ)
18	14, 17	syl6bb 252	. . . . . . 7 ⊢ (z = y → ([z / x]φ ↔ ψ))
19		equequ1 1684	. . . . . . 7 ⊢ (z = y → (z = w ↔ y = w))
20	18, 19	bibi12d 312	. . . . . 6 ⊢ (z = y → (([z / x]φ ↔ z = w) ↔ (ψ ↔ y = w)))
21	12, 13, 20	cbval 1984	. . . . 5 ⊢ (∀z([z / x]φ ↔ z = w) ↔ ∀y(ψ ↔ y = w))
22	8, 21	bitri 240	. . . 4 ⊢ (∀x(φ ↔ x = w) ↔ ∀y(ψ ↔ y = w))
23	22	abbii 2466	. . 3 ⊢ {w ∣ ∀x(φ ↔ x = w)} = {w ∣ ∀y(ψ ↔ y = w)}
24	23	unieqi 3902	. 2 ⊢ ∪{w ∣ ∀x(φ ↔ x = w)} = ∪{w ∣ ∀y(ψ ↔ y = w)}
25		dfiota2 4341	. 2 ⊢ (℩xφ) = ∪{w ∣ ∀x(φ ↔ x = w)}
26		dfiota2 4341	. 2 ⊢ (℩yψ) = ∪{w ∣ ∀y(ψ ↔ y = w)}
27	24, 25, 26	3eqtr4i 2383	1 ⊢ (℩xφ) = (℩yψ)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  [wsb 1648  {cab 2339  ∪cuni 3892  ℩cio 4338

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 2479  df-rex 2621  df-sn 3742  df-uni 3893  df-iota 4340

This theorem is referenced by:  cbviotav  4346