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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 3060 | . 2 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑)) | |
2 | vex 3482 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | a1bi 362 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V → 𝜑)) |
4 | 3 | albii 1816 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑)) |
5 | 1, 4 | bitr4i 278 | 1 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 |
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-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-v 3480 |
This theorem is referenced by: viin 5070 ralcom4f 32496 hfext 36165 clsk1independent 44036 ntrneiel2 44076 ntrneik4w 44090 |
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