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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1441g | Structured version Visualization version GIF version | ||
| Description: First-order logic and set theory. See bnj1441 34854 for a version with more disjoint variable conditions, but not requiring ax-13 2377. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj1441g.1 | ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴) |
| bnj1441g.2 | ⊢ (𝜑 → ∀𝑦𝜑) |
| Ref | Expression |
|---|---|
| bnj1441g | ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-rab 3437 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
| 2 | bnj1441g.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴) | |
| 3 | bnj1441g.2 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) | |
| 4 | 2, 3 | hban 2300 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∀𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)) |
| 5 | 4 | hbabg 2726 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
| 6 | 1, 5 | hbxfreq 2872 | 1 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 {cab 2714 {crab 3436 |
| 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-13 2377 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 |
| This theorem is referenced by: (None) |
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