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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 3202 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
2 | r19.3rzv 4402 | . . 3 ⊢ (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
3 | 2 | con1bid 359 | . 2 ⊢ (𝐴 ≠ ∅ → (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝜑)) |
4 | 1, 3 | syl5rbb 287 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-rex 3112 df-dif 3884 df-nul 4244 |
This theorem is referenced by: r19.45zv 4406 r19.44zv 4407 r19.36zv 4410 iunconst 4890 lcmgcdlem 15940 pmtrprfvalrn 18608 dvdsr02 19402 voliune 31598 dya2iocuni 31651 filnetlem4 33842 prmunb2 41015 |
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