![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 1912 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.3rz 4503 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ≠ wne 2938 ∀wral 3059 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-ne 2939 df-ral 3060 df-dif 3966 df-nul 4340 |
This theorem is referenced by: r19.9rzv 4506 r19.37zv 4508 ralnralall 4521 iinconst 5007 cnvpo 6309 supicc 13538 coe1mul2lem1 22286 neipeltop 23153 utop3cls 24276 tgcgr4 28554 frgrregord013 30424 poimirlem23 37630 rencldnfi 42809 cvgdvgrat 44309 |
Copyright terms: Public domain | W3C validator |