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Mirrors > Home > NFE Home > Th. List > iota5 | GIF version |
Description: A method for computing iota. (Contributed by NM, 17-Sep-2013.) |
Ref | Expression |
---|---|
iota5.1 | ⊢ ((φ ∧ A ∈ V) → (ψ ↔ x = A)) |
Ref | Expression |
---|---|
iota5 | ⊢ ((φ ∧ A ∈ V) → (℩xψ) = A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iota5.1 | . . 3 ⊢ ((φ ∧ A ∈ V) → (ψ ↔ x = A)) | |
2 | 1 | alrimiv 1631 | . 2 ⊢ ((φ ∧ A ∈ V) → ∀x(ψ ↔ x = A)) |
3 | eqeq2 2362 | . . . . . . 7 ⊢ (y = A → (x = y ↔ x = A)) | |
4 | 3 | bibi2d 309 | . . . . . 6 ⊢ (y = A → ((ψ ↔ x = y) ↔ (ψ ↔ x = A))) |
5 | 4 | albidv 1625 | . . . . 5 ⊢ (y = A → (∀x(ψ ↔ x = y) ↔ ∀x(ψ ↔ x = A))) |
6 | eqeq2 2362 | . . . . 5 ⊢ (y = A → ((℩xψ) = y ↔ (℩xψ) = A)) | |
7 | 5, 6 | imbi12d 311 | . . . 4 ⊢ (y = A → ((∀x(ψ ↔ x = y) → (℩xψ) = y) ↔ (∀x(ψ ↔ x = A) → (℩xψ) = A))) |
8 | iotaval 4350 | . . . 4 ⊢ (∀x(ψ ↔ x = y) → (℩xψ) = y) | |
9 | 7, 8 | vtoclg 2914 | . . 3 ⊢ (A ∈ V → (∀x(ψ ↔ x = A) → (℩xψ) = A)) |
10 | 9 | adantl 452 | . 2 ⊢ ((φ ∧ A ∈ V) → (∀x(ψ ↔ x = A) → (℩xψ) = A)) |
11 | 2, 10 | mpd 14 | 1 ⊢ ((φ ∧ A ∈ V) → (℩xψ) = A) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 = wceq 1642 ∈ wcel 1710 ℩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-nan 1288 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-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: (None) |
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