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Mirrors > Home > ILE Home > Th. List > Mathboxes > elabgf2 | GIF version |
Description: One implication of elabgf 2821. (Contributed by BJ, 21-Nov-2019.) |
Ref | Expression |
---|---|
elabgf2.nf1 | ⊢ Ⅎ𝑥𝐴 |
elabgf2.nf2 | ⊢ Ⅎ𝑥𝜓 |
elabgf2.1 | ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) |
Ref | Expression |
---|---|
elabgf2 | ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elabgf2.nf1 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | elabgf2.nf2 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | nfab1 2281 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
4 | 1, 3 | nfel 2288 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑} |
5 | 2, 4 | nfim 1551 | . 2 ⊢ Ⅎ𝑥(𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
6 | elabgf0 12973 | . 2 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) | |
7 | bicom1 130 | . . 3 ⊢ ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
8 | elabgf2.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) | |
9 | bi1 117 | . . . 4 ⊢ ((𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) → (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
10 | 8, 9 | syl9 72 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))) |
11 | 7, 10 | syl5 32 | . 2 ⊢ (𝑥 = 𝐴 → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))) |
12 | 1, 5, 6, 11 | bj-vtoclgf 12972 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 Ⅎwnf 1436 ∈ wcel 1480 {cab 2123 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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 |
This theorem is referenced by: elabf2 12978 elabg2 12981 bj-intabssel1 12986 |
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