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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 3134 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑) | |
2 | r19.3rzv 4208 | . . 3 ⊢ (𝐴 ≠ ∅ → (¬ 𝜑 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜑)) | |
3 | 2 | con1bid 344 | . 2 ⊢ (𝐴 ≠ ∅ → (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝜑)) |
4 | 1, 3 | syl5rbb 273 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ≠ wne 2932 ∀wral 3050 ∃wrex 3051 ∅c0 4058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-v 3342 df-dif 3718 df-nul 4059 |
This theorem is referenced by: r19.45zv 4212 r19.44zv 4213 r19.36zv 4216 iunconst 4681 lcmgcdlem 15521 pmtrprfvalrn 18108 dvdsr02 18856 voliune 30601 dya2iocuni 30654 filnetlem4 32682 prmunb2 39012 |
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