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Mirrors > Home > MPE Home > Th. List > r19.3rzv | Structured version Visualization version GIF version |
Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.) |
Ref | Expression |
---|---|
r19.3rzv | ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1910 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.3rz 4501 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ≠ wne 2930 ∀wral 3051 ∅c0 4325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-9 2109 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-ne 2931 df-ral 3052 df-dif 3950 df-nul 4326 |
This theorem is referenced by: r19.9rzv 4504 r19.37zv 4506 ralnralall 4523 iinconst 5013 cnvpo 6300 supicc 13534 coe1mul2lem1 22260 neipeltop 23127 utop3cls 24250 tgcgr4 28461 frgrregord013 30331 poimirlem23 37346 rencldnfi 42496 cvgdvgrat 44005 |
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