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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1317 | 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 |
---|---|
bnj1317.1 | ⊢ 𝐴 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
bnj1317 | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1317.1 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝜑} | |
2 | hbab1 2807 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | hbxfreq 2942 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 = wceq 1533 ∈ wcel 2110 {cab 2799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 |
This theorem is referenced by: bnj1014 32233 bnj1145 32265 bnj1384 32304 bnj1398 32306 bnj1448 32319 bnj1450 32322 bnj1466 32325 bnj1463 32327 bnj1491 32329 bnj1497 32332 bnj1498 32333 bnj1520 32338 bnj1501 32339 |
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