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Mirrors > Home > NFE Home > Th. List > iota2df | GIF version |
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.) |
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) |
Ref | Expression |
---|---|
iota2df | ⊢ (φ → (χ ↔ (℩xψ) = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iota2df.6 | . 2 ⊢ (φ → ℲxB) | |
2 | iota2df.5 | . . 3 ⊢ (φ → Ⅎxχ) | |
3 | nfiota1 4341 | . . . . 5 ⊢ Ⅎx(℩xψ) | |
4 | 3 | a1i 10 | . . . 4 ⊢ (φ → Ⅎx(℩xψ)) |
5 | 4, 1 | nfeqd 2503 | . . 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 4353 | . . . 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 2905 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 Ⅎwnf 1544 = wceq 1642 ∈ wcel 1710 ∃!weu 2204 Ⅎwnfc 2476 ℩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-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 2478 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 df-uni 3892 df-iota 4339 |
This theorem is referenced by: iota2d 4366 iota2 4367 |
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