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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 1914 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | 1 | r19.3rz 4497 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ≠ wne 2940 ∀wral 3061 ∅c0 4333 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-ne 2941 df-ral 3062 df-dif 3954 df-nul 4334 | 
| This theorem is referenced by: r19.9rzv 4500 r19.37zv 4502 ralnralall 4515 iinconst 5002 cnvpo 6307 supicc 13541 coe1mul2lem1 22270 neipeltop 23137 utop3cls 24260 tgcgr4 28539 frgrregord013 30414 poimirlem23 37650 rencldnfi 42832 cvgdvgrat 44332 | 
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