Theorem iota2df 4366

Description: A condition that allows us to represent "the unique element such that φ " with a class expression A. (Contributed by NM, 30-Dec-2014.)

Hypotheses

Ref	Expression
iota2df.1	⊢ (φ → B ∈ V)
iota2df.2	⊢ (φ → ∃!xψ)
iota2df.3	⊢ ((φ ∧ x = B) → (ψ ↔ χ))
iota2df.4	⊢ Ⅎxφ
iota2df.5	⊢ (φ → Ⅎxχ)
iota2df.6	⊢ (φ → ℲxB)

Assertion

Ref	Expression
iota2df	⊢ (φ → (χ ↔ (℩xψ) = B))

Proof of Theorem iota2df

Step	Hyp	Ref	Expression
1		iota2df.6	. 2 ⊢ (φ → ℲxB)
2		iota2df.5	. . 3 ⊢ (φ → Ⅎxχ)
3		nfiota1 4342	. . . . 5 ⊢ Ⅎx(℩xψ)
4	3	a1i 10	. . . 4 ⊢ (φ → Ⅎx(℩xψ))
5	4, 1	nfeqd 2504	. . 3 ⊢ (φ → Ⅎx(℩xψ) = B)
6	2, 5	nfbid 1832	. 2 ⊢ (φ → Ⅎx(χ ↔ (℩xψ) = B))
7		iota2df.4	. . 3 ⊢ Ⅎxφ
8		iota2df.3	. . . . 5 ⊢ ((φ ∧ x = B) → (ψ ↔ χ))
9		simpr 447	. . . . . 6 ⊢ ((φ ∧ x = B) → x = B)
10	9	eqeq2d 2364	. . . . 5 ⊢ ((φ ∧ x = B) → ((℩xψ) = x ↔ (℩xψ) = B))
11	8, 10	bibi12d 312	. . . 4 ⊢ ((φ ∧ x = B) → ((ψ ↔ (℩xψ) = x) ↔ (χ ↔ (℩xψ) = B)))
12	11	ex 423	. . 3 ⊢ (φ → (x = B → ((ψ ↔ (℩xψ) = x) ↔ (χ ↔ (℩xψ) = B))))
13	7, 12	alrimi 1765	. 2 ⊢ (φ → ∀x(x = B → ((ψ ↔ (℩xψ) = x) ↔ (χ ↔ (℩xψ) = B))))
14		iota2df.2	. . . 4 ⊢ (φ → ∃!xψ)
15		iota1 4354	. . . 4 ⊢ (∃!xψ → (ψ ↔ (℩xψ) = x))
16	14, 15	syl 15	. . 3 ⊢ (φ → (ψ ↔ (℩xψ) = x))
17	7, 16	alrimi 1765	. 2 ⊢ (φ → ∀x(ψ ↔ (℩xψ) = x))
18		iota2df.1	. 2 ⊢ (φ → B ∈ V)
19		vtoclgft 2906	. 2 ⊢ (((ℲxB ∧ Ⅎx(χ ↔ (℩xψ) = B)) ∧ (∀x(x = B → ((ψ ↔ (℩xψ) = x) ↔ (χ ↔ (℩xψ) = B))) ∧ ∀x(ψ ↔ (℩xψ) = x)) ∧ B ∈ V) → (χ ↔ (℩xψ) = B))
20	1, 6, 13, 17, 18, 19	syl221anc 1193	1 ⊢ (φ → (χ ↔ (℩xψ) = B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Ⅎwnfc 2477  ℩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-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by:  iota2d  4367  iota2  4368