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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1436 | 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 |
---|---|
bnj1436.1 | ⊢ 𝐴 = {𝑥 ∣ 𝜑} |
Ref | Expression |
---|---|
bnj1436 | ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1436.1 | . . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑} | |
2 | 1 | abeq2i 2912 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) |
3 | 2 | biimpi 208 | 1 ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 {cab 2785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-tru 1657 df-ex 1876 df-sb 2065 df-clab 2786 df-cleq 2792 df-clel 2795 |
This theorem is referenced by: bnj1517 31437 bnj66 31447 bnj900 31516 bnj1296 31606 bnj1371 31614 bnj1374 31616 bnj1398 31619 bnj1450 31635 bnj1497 31645 bnj1498 31646 bnj1514 31648 bnj1501 31652 |
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