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Mirrors > Home > ILE Home > Th. List > Mathboxes > elabgft1 | GIF version |
Description: One implication of elabgf 2872, in closed form. (Contributed by BJ, 21-Nov-2019.) |
Ref | Expression |
---|---|
elabgf1.nf1 | ⊢ Ⅎ𝑥𝐴 |
elabgf1.nf2 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
elabgft1 | ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biimp 117 | . . . . . 6 ⊢ ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜑)) | |
2 | imim2 55 | . . . . . 6 ⊢ ((𝜑 → 𝜓) → ((𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))) | |
3 | 1, 2 | syl5 32 | . . . . 5 ⊢ ((𝜑 → 𝜓) → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))) |
4 | 3 | imim2i 12 | . . . 4 ⊢ ((𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝑥 = 𝐴 → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)))) |
5 | 4 | alimi 1448 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)))) |
6 | elabgf1.nf1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
7 | nfab1 2314 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
8 | 6, 7 | nfel 2321 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑} |
9 | elabgf1.nf2 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
10 | 8, 9 | nfim 1565 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓) |
11 | elabgf0 13777 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)) | |
12 | 6, 10, 11 | bj-vtoclgft 13775 | . . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))) |
13 | 5, 12 | syl 14 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))) |
14 | 13 | pm2.43d 50 | 1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1346 = wceq 1348 Ⅎwnf 1453 ∈ wcel 2141 {cab 2156 Ⅎwnfc 2299 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 |
This theorem is referenced by: elabgf1 13779 |
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