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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1441 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Add disjoint variable condition to avoid ax-13 2365. See bnj1441g 34380 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1441.1 | ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴) |
bnj1441.2 | ⊢ (𝜑 → ∀𝑦𝜑) |
Ref | Expression |
---|---|
bnj1441 | ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3427 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | bnj1441.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴) | |
3 | bnj1441.2 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) | |
4 | 2, 3 | hban 2288 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∀𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)) |
5 | 4 | hbab 2714 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
6 | 1, 5 | hbxfreq 2858 | 1 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1531 ∈ wcel 2098 {cab 2703 {crab 3426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-rab 3427 |
This theorem is referenced by: (None) |
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