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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 1883 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.3rz 4095 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ≠ wne 2823 ∀wral 2941 ∅c0 3948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-v 3233 df-dif 3610 df-nul 3949 |
This theorem is referenced by: r19.9rzv 4098 r19.37zv 4100 ralnralall 4113 iinconst 4562 cnvpo 5711 supicc 12358 coe1mul2lem1 19685 neipeltop 20981 utop3cls 22102 tgcgr4 25471 frgrregord013 27382 poimirlem23 33562 rencldnfi 37702 cvgdvgrat 38829 |
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