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Mirrors > Home > ILE Home > Th. List > alral | GIF version |
Description: Universal quantification implies restricted quantification. (Contributed by NM, 20-Oct-2006.) |
Ref | Expression |
---|---|
alral | ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 5 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜑)) | |
2 | 1 | alimi 1384 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) |
3 | df-ral 2353 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
4 | 2, 3 | sylibr 132 | 1 ⊢ (∀𝑥𝜑 → ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 ∈ wcel 1433 ∀wral 2348 |
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 |
This theorem depends on definitions: df-bi 115 df-ral 2353 |
This theorem is referenced by: find 4340 findset 10740 |
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