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| Mirrors > Home > MPE Home > Th. List > ra4v | Structured version Visualization version GIF version | ||
| Description: Version of ra4 3886 with a disjoint variable condition, requiring fewer axioms. This is stdpc5v 1938 for a restricted domain. (Contributed by BJ, 27-Mar-2020.) | 
| Ref | Expression | 
|---|---|
| ra4v | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | r19.21v 3180 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | |
| 2 | 1 | biimpi 216 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wral 3061 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ral 3062 | 
| This theorem is referenced by: wfr3g 8347 frr3g 9796 | 
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