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Mirrors > Home > ILE Home > Th. List > ralv | 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 2419 | . 2 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑)) | |
2 | vex 2684 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | a1bi 242 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V → 𝜑)) |
4 | 3 | albii 1446 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑)) |
5 | 1, 4 | bitr4i 186 | 1 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1329 ∈ wcel 1480 ∀wral 2414 Vcvv 2681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-ral 2419 df-v 2683 |
This theorem is referenced by: ralcom4 2703 viin 3867 issref 4916 frecrdg 6298 |
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