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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 1892 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.3rz 4356 | 1 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ≠ wne 2984 ∀wral 3105 ∅c0 4211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-dif 3862 df-nul 4212 |
This theorem is referenced by: r19.9rzv 4359 r19.37zv 4361 ralnralall 4372 iinconst 4835 cnvpo 6013 supicc 12736 coe1mul2lem1 20118 neipeltop 21421 utop3cls 22543 tgcgr4 25999 frgrregord013 27866 poimirlem23 34446 rencldnfi 38903 cvgdvgrat 40183 |
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