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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1476 | Structured version Visualization version GIF version | ||
| Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj1476.1 | ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} |
| bnj1476.2 | ⊢ (𝜓 → 𝐷 = ∅) |
| Ref | Expression |
|---|---|
| bnj1476 | ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1476.2 | . . . 4 ⊢ (𝜓 → 𝐷 = ∅) | |
| 2 | bnj1476.1 | . . . . . 6 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} | |
| 3 | nfrab1 3457 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 𝜑} | |
| 4 | 2, 3 | nfcxfr 2903 | . . . . 5 ⊢ Ⅎ𝑥𝐷 |
| 5 | 4 | eq0f 4347 | . . . 4 ⊢ (𝐷 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐷) |
| 6 | 1, 5 | sylib 218 | . . 3 ⊢ (𝜓 → ∀𝑥 ¬ 𝑥 ∈ 𝐷) |
| 7 | 2 | reqabi 3460 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)) |
| 8 | 7 | notbii 320 | . . . 4 ⊢ (¬ 𝑥 ∈ 𝐷 ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)) |
| 9 | iman 401 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)) | |
| 10 | 8, 9 | sylbb2 238 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐷 → (𝑥 ∈ 𝐴 → 𝜑)) |
| 11 | 6, 10 | sylg 1823 | . 2 ⊢ (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
| 12 | 11 | bnj1142 34803 | 1 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rab 3437 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: bnj1312 35072 |
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