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Mirrors > Home > NFE Home > Th. List > nfeld | GIF version |
Description: Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
nfeqd.1 | ⊢ (φ → ℲxA) |
nfeqd.2 | ⊢ (φ → ℲxB) |
Ref | Expression |
---|---|
nfeld | ⊢ (φ → Ⅎx A ∈ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clel 2349 | . 2 ⊢ (A ∈ B ↔ ∃y(y = A ∧ y ∈ B)) | |
2 | nfv 1619 | . . 3 ⊢ Ⅎyφ | |
3 | nfcvd 2490 | . . . . 5 ⊢ (φ → Ⅎxy) | |
4 | nfeqd.1 | . . . . 5 ⊢ (φ → ℲxA) | |
5 | 3, 4 | nfeqd 2503 | . . . 4 ⊢ (φ → Ⅎx y = A) |
6 | nfeqd.2 | . . . . 5 ⊢ (φ → ℲxB) | |
7 | 6 | nfcrd 2502 | . . . 4 ⊢ (φ → Ⅎx y ∈ B) |
8 | 5, 7 | nfand 1822 | . . 3 ⊢ (φ → Ⅎx(y = A ∧ y ∈ B)) |
9 | 2, 8 | nfexd 1854 | . 2 ⊢ (φ → Ⅎx∃y(y = A ∧ y ∈ B)) |
10 | 1, 9 | nfxfrd 1571 | 1 ⊢ (φ → Ⅎx A ∈ B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∃wex 1541 Ⅎwnf 1544 = wceq 1642 ∈ wcel 1710 Ⅎwnfc 2476 |
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-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-nf 1545 df-cleq 2346 df-clel 2349 df-nfc 2478 |
This theorem is referenced by: nfneld 2613 nfrald 2665 ralcom2 2775 nfreud 2783 nfrmod 2784 nfrmo 2786 nfsbc1d 3063 nfsbcd 3066 sbcrext 3119 nfbrd 4682 |
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