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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ralimralim | Structured version Visualization version GIF version |
Description: Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
ralimralim | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra1 3122 | . 2 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | |
2 | rspa 3111 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜑) | |
3 | ax-1 6 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜑)) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜑)) |
5 | 4 | ex 403 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑))) |
6 | 1, 5 | ralrimi 3138 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2106 ∀wral 3089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-10 2134 ax-12 2162 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-ex 1824 df-nf 1828 df-ral 3094 |
This theorem is referenced by: infxrunb2 40485 |
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