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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 2487 | . 2 ⊢ (∃!𝑥𝜑 → ∃*𝑥𝜑) | |
2 | moabex 4854 | . 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 1977 ∃!weu 2458 ∃*wmo 2459 {cab 2596 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 df-pr 4128 |
This theorem is referenced by: (None) |
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