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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 2979 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
2 | r19.3rzv 4016 | . . 3 ⊢ (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
3 | 2 | con1bid 344 | . 2 ⊢ (𝐴 ≠ ∅ → (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝜑)) |
4 | 1, 3 | syl5rbb 272 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∅c0 3874 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-v 3175 df-dif 3543 df-nul 3875 |
This theorem is referenced by: r19.45zv 4020 r19.44zv 4021 r19.36zv 4024 iunconst 4465 lcmgcdlem 15157 pmtrprfvalrn 17731 dvdsr02 18479 voliune 29619 dya2iocuni 29672 filnetlem4 31546 prmunb2 37532 |
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