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Mirrors > Home > MPE Home > Th. List > elabd | Structured version Visualization version GIF version |
Description: Explicit demonstration the class {𝑥 ∣ 𝜓} is not empty by the example 𝑋. (Contributed by RP, 12-Aug-2020.) |
Ref | Expression |
---|---|
elab.xex | ⊢ (𝜑 → 𝑋 ∈ V) |
elab.xmaj | ⊢ (𝜑 → 𝜒) |
elab.xsub | ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
elabd | ⊢ (𝜑 → ∃𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elab.xex | . 2 ⊢ (𝜑 → 𝑋 ∈ V) | |
2 | elab.xmaj | . 2 ⊢ (𝜑 → 𝜒) | |
3 | elab.xsub | . . 3 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜒)) | |
4 | 3 | spcegv 3443 | . 2 ⊢ (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓)) |
5 | 1, 2, 4 | sylc 65 | 1 ⊢ (𝜑 → ∃𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1630 ∃wex 1851 ∈ wcel 2144 Vcvv 3349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-v 3351 |
This theorem is referenced by: hasheqf1od 13345 setsexstruct2 16103 wwlksnextbij 27044 clrellem 38448 clcnvlem 38449 uspgrsprfo 42274 uspgrbispr 42277 |
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