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Mirrors > Home > MPE Home > Th. List > r19.9rzv | Structured version Visualization version GIF version |
Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.) |
Ref | Expression |
---|---|
r19.9rzv | ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrex2 3071 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
2 | r19.3rzv 4505 | . . 3 ⊢ (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
3 | 2 | con1bid 355 | . 2 ⊢ (𝐴 ≠ ∅ → (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝜑)) |
4 | 1, 3 | bitr2id 284 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-ne 2939 df-ral 3060 df-rex 3069 df-dif 3966 df-nul 4340 |
This theorem is referenced by: r19.45zv 4509 r19.44zv 4510 r19.36zv 4513 iunconst 5006 lcmgcdlem 16640 pmtrprfvalrn 19521 dvdsr02 20389 voliune 34210 dya2iocuni 34265 filnetlem4 36364 prmunb2 44307 |
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