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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 3316 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 𝜑} | |
4 | 2, 3 | nfcxfr 2905 | . . . . 5 ⊢ Ⅎ𝑥𝐷 |
5 | 4 | eq0f 4276 | . . . 4 ⊢ (𝐷 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐷) |
6 | 1, 5 | sylib 217 | . . 3 ⊢ (𝜓 → ∀𝑥 ¬ 𝑥 ∈ 𝐷) |
7 | 2 | rabeq2i 3421 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)) |
8 | 7 | notbii 320 | . . . 4 ⊢ (¬ 𝑥 ∈ 𝐷 ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)) |
9 | iman 402 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)) | |
10 | 8, 9 | sylbb2 237 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐷 → (𝑥 ∈ 𝐴 → 𝜑)) |
11 | 6, 10 | sylg 1825 | . 2 ⊢ (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
12 | 11 | bnj1142 32756 | 1 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀wal 1537 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 ∅c0 4258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rab 3073 df-dif 3891 df-nul 4259 |
This theorem is referenced by: bnj1312 33025 |
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