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| Mirrors > Home > MPE Home > Th. List > r19.21 | Structured version Visualization version GIF version | ||
| Description: Restricted quantifier version of 19.21 2207. (Contributed by Scott Fenton, 30-Mar-2011.) | 
| Ref | Expression | 
|---|---|
| r19.21.1 | ⊢ Ⅎ𝑥𝜑 | 
| Ref | Expression | 
|---|---|
| r19.21 | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | r19.21.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | r19.21t 3253 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 Ⅎwnf 1783 ∀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 ax-6 1967 ax-7 2007 ax-12 2177 | 
| This theorem depends on definitions: df-bi 207 df-ex 1780 df-nf 1784 df-ral 3062 | 
| This theorem is referenced by: rmo3f 3740 ra4 3886 rmoanim 3894 rmoanimALT 3895 r19.32 47110 | 
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