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Mirrors > Home > MPE Home > Th. List > euabex | Structured version Visualization version GIF version |
Description: The abstraction of a wff with existential uniqueness exists. (Contributed by NM, 25-Nov-1994.) |
Ref | Expression |
---|---|
euabex | ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eumo 2578 | . 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
2 | moabex 5374 | . 2 ⊢ (∃*𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) | |
3 | 1, 2 | syl 17 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∃*wmo 2538 ∃!weu 2568 {cab 2715 Vcvv 3431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-rab 3073 df-v 3433 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-sn 4564 df-pr 4566 |
This theorem is referenced by: sprval 44888 prprval 44923 |
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