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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 2925 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) |
3 | 2 | biimpi 219 | 1 ⊢ (𝑥 ∈ 𝐴 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {cab 2776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 |
This theorem is referenced by: bnj1517 32232 bnj66 32242 bnj900 32311 bnj1296 32403 bnj1371 32411 bnj1374 32413 bnj1398 32416 bnj1450 32432 bnj1497 32442 bnj1498 32443 bnj1514 32445 bnj1501 32449 |
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