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Mirrors > Home > NFE Home > Th. List > nfreud | GIF version |
Description: Deduction version of nfreu 2785. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.) |
Ref | Expression |
---|---|
nfreud.1 | ⊢ Ⅎyφ |
nfreud.2 | ⊢ (φ → ℲxA) |
nfreud.3 | ⊢ (φ → Ⅎxψ) |
Ref | Expression |
---|---|
nfreud | ⊢ (φ → Ⅎx∃!y ∈ A ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 2621 | . 2 ⊢ (∃!y ∈ A ψ ↔ ∃!y(y ∈ A ∧ ψ)) | |
2 | nfreud.1 | . . 3 ⊢ Ⅎyφ | |
3 | nfcvf 2511 | . . . . . 6 ⊢ (¬ ∀x x = y → Ⅎxy) | |
4 | 3 | adantl 452 | . . . . 5 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxy) |
5 | nfreud.2 | . . . . . 6 ⊢ (φ → ℲxA) | |
6 | 5 | adantr 451 | . . . . 5 ⊢ ((φ ∧ ¬ ∀x x = y) → ℲxA) |
7 | 4, 6 | nfeld 2504 | . . . 4 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx y ∈ A) |
8 | nfreud.3 | . . . . 5 ⊢ (φ → Ⅎxψ) | |
9 | 8 | adantr 451 | . . . 4 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ) |
10 | 7, 9 | nfand 1822 | . . 3 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎx(y ∈ A ∧ ψ)) |
11 | 2, 10 | nfeud2 2216 | . 2 ⊢ (φ → Ⅎx∃!y(y ∈ A ∧ ψ)) |
12 | 1, 11 | nfxfrd 1571 | 1 ⊢ (φ → Ⅎx∃!y ∈ A ψ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 358 ∀wal 1540 Ⅎwnf 1544 = wceq 1642 ∈ wcel 1710 ∃!weu 2204 Ⅎwnfc 2476 ∃!wreu 2616 |
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-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-cleq 2346 df-clel 2349 df-nfc 2478 df-reu 2621 |
This theorem is referenced by: nfreu 2785 |
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