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Mirrors > Home > ILE 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 2133 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | nfv 1508 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvd 2280 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) | |
4 | nfeqd.1 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 3, 4 | nfeqd 2294 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝐴) |
6 | nfeqd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
7 | 6 | nfcrd 2293 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
8 | 5, 7 | nfand 1547 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) |
9 | 2, 8 | nfexd 1734 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) |
10 | 1, 9 | nfxfrd 1451 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 Ⅎwnf 1436 ∃wex 1468 ∈ wcel 1480 Ⅎwnfc 2266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-cleq 2130 df-clel 2133 df-nfc 2268 |
This theorem is referenced by: nfneld 2409 nfraldxy 2465 nfrexdxy 2466 nfreudxy 2602 nfsbc1d 2920 nfsbcd 2923 sbcrext 2981 nfbrd 3968 nfriotadxy 5731 nfixpxy 6604 |
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