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Mirrors > Home > MPE Home > Th. List > ralv | Structured version Visualization version GIF version |
Description: A universal quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.) |
Ref | Expression |
---|---|
ralv | ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3087 | . 2 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑)) | |
2 | vex 3412 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | a1bi 355 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V → 𝜑)) |
4 | 3 | albii 1782 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑)) |
5 | 1, 4 | bitr4i 270 | 1 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1505 ∈ wcel 2050 ∀wral 3082 Vcvv 3409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-ext 2744 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-ral 3087 df-v 3411 |
This theorem is referenced by: ralcom4OLD 3440 viin 4848 ralcom4f 30009 hfext 33165 clsk1independent 39759 ntrneiel2 39799 ntrneik4w 39813 |
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