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Mirrors > Home > ILE Home > Th. List > Mathboxes > elabgf2 | GIF version |
Description: One implication of elabgf 2685. (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 2180 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
4 | 1, 3 | nfel 2186 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑} |
5 | 2, 4 | nfim 1464 | . 2 ⊢ Ⅎ𝑥(𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
6 | elabgf0 9916 | . 2 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) | |
7 | bicom1 122 | . . 3 ⊢ ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
8 | elabgf2.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑)) | |
9 | bi1 111 | . . . 4 ⊢ ((𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) → (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜑})) | |
10 | 8, 9 | syl9 66 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))) |
11 | 7, 10 | syl5 28 | . 2 ⊢ (𝑥 = 𝐴 → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))) |
12 | 1, 5, 6, 11 | bj-vtoclgf 9915 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 {cab 2026 Ⅎwnfc 2165 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 |
This theorem is referenced by: elabf2 9921 elabg2 9924 bj-intabssel1 9929 |
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