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Mirrors > Home > ILE Home > Th. List > Mathboxes > elabgf2 | GIF version |
Description: One implication of elabgf 2854. (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 2301 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
4 | 1, 3 | nfel 2308 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑} |
5 | 2, 4 | nfim 1552 | . 2 ⊢ Ⅎ𝑥(𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}) |
6 | elabgf0 13393 | . 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 | biimp 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 13392 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1335 Ⅎwnf 1440 ∈ wcel 2128 {cab 2143 Ⅎwnfc 2286 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 |
This theorem is referenced by: elabf2 13398 elabg2 13401 bj-intabssel1 13406 |
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