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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 3062 | . 2 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑)) | |
| 2 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | 2 | a1bi 362 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V → 𝜑)) |
| 4 | 3 | albii 1819 | . 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 1538 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-v 3482 |
| This theorem is referenced by: viin 5065 ralcom4f 32486 hfext 36184 clsk1independent 44059 ntrneiel2 44099 ntrneik4w 44113 |
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