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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 5464 | . 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 2108 ∃*wmo 2538 ∃!weu 2568 {cab 2714 Vcvv 3480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-un 3956 df-in 3958 df-ss 3968 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: tfsconcatun 43350 sprval 47466 prprval 47501 |
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