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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 32131 for a version with more disjoint variable conditions, but not requiring ax-13 2389. (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 3146 | . 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 2307 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∀𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)) |
5 | 4 | hbabg 2810 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
6 | 1, 5 | hbxfreq 2941 | 1 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1534 ∈ wcel 2113 {cab 2798 {crab 3141 |
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-13 2389 ax-ext 2792 |
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 2799 df-cleq 2813 df-clel 2892 df-rab 3146 |
This theorem is referenced by: (None) |
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