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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 3169 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
2 | r19.3rzv 4435 | . . 3 ⊢ (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
3 | 2 | con1bid 356 | . 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 205 ≠ wne 2945 ∀wral 3066 ∃wrex 3067 ∅c0 4262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-9 2120 ax-12 2175 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-clab 2718 df-cleq 2732 df-ne 2946 df-ral 3071 df-rex 3072 df-dif 3895 df-nul 4263 |
This theorem is referenced by: r19.45zv 4439 r19.44zv 4440 r19.36zv 4443 iunconst 4939 lcmgcdlem 16307 pmtrprfvalrn 19092 dvdsr02 19894 voliune 32191 dya2iocuni 32244 filnetlem4 34564 prmunb2 41897 |
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