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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 3387 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 𝜑} | |
4 | 2, 3 | nfcxfr 2978 | . . . . 5 ⊢ Ⅎ𝑥𝐷 |
5 | 4 | eq0f 4308 | . . . 4 ⊢ (𝐷 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐷) |
6 | 1, 5 | sylib 220 | . . 3 ⊢ (𝜓 → ∀𝑥 ¬ 𝑥 ∈ 𝐷) |
7 | 2 | rabeq2i 3490 | . . . . 5 ⊢ (𝑥 ∈ 𝐷 ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)) |
8 | 7 | notbii 322 | . . . 4 ⊢ (¬ 𝑥 ∈ 𝐷 ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)) |
9 | iman 404 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑)) | |
10 | 8, 9 | sylbb2 240 | . . 3 ⊢ (¬ 𝑥 ∈ 𝐷 → (𝑥 ∈ 𝐴 → 𝜑)) |
11 | 6, 10 | sylg 1822 | . 2 ⊢ (𝜓 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
12 | 11 | bnj1142 32065 | 1 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1534 = wceq 1536 ∈ wcel 2113 ∀wral 3141 {crab 3145 ∅c0 4294 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-dif 3942 df-nul 4295 |
This theorem is referenced by: bnj1312 32334 |
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