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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 2173 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | nfv 1528 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvd 2320 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) | |
4 | nfeqd.1 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 3, 4 | nfeqd 2334 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 = 𝐴) |
6 | nfeqd.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
7 | 6 | nfcrd 2333 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐵) |
8 | 5, 7 | nfand 1568 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) |
9 | 2, 8 | nfexd 1761 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑦(𝑦 = 𝐴 ∧ 𝑦 ∈ 𝐵)) |
10 | 1, 9 | nfxfrd 1475 | 1 ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 Ⅎwnf 1460 ∃wex 1492 ∈ wcel 2148 Ⅎwnfc 2306 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-17 1526 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-cleq 2170 df-clel 2173 df-nfc 2308 |
This theorem is referenced by: nfneld 2450 nfraldw 2509 nfraldxy 2510 nfrexdxy 2511 nfreudxy 2650 nfsbc1d 2979 nfsbcd 2982 sbcrext 3040 nfsbcdw 3091 nfbrd 4047 nfriotadxy 5836 nfixpxy 6714 |
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