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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 2353 | . 2 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑)) | |
2 | vex 2604 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | a1bi 241 | . . 3 ⊢ (𝜑 ↔ (𝑥 ∈ V → 𝜑)) |
4 | 3 | albii 1399 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ V → 𝜑)) |
5 | 1, 4 | bitr4i 185 | 1 ⊢ (∀𝑥 ∈ V 𝜑 ↔ ∀𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 ∈ wcel 1433 ∀wral 2348 Vcvv 2601 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-ral 2353 df-v 2603 |
This theorem is referenced by: ralcom4 2621 viin 3737 issref 4727 |
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