|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > r19.28v | Structured version Visualization version GIF version | ||
| Description: Restricted quantifier version of one direction of 19.28 2227. (Assuming Ⅎ𝑥𝐴, the other direction holds when 𝐴 is nonempty, see r19.28zv 4500.) (Contributed by NM, 2-Apr-2004.) (Proof shortened by Wolf Lammen, 17-Jun-2023.) | 
| Ref | Expression | 
|---|---|
| r19.28v | ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . . . 4 ⊢ (𝜑 → 𝜑) | |
| 2 | 1 | ralrimivw 3149 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑) | 
| 3 | 2 | anim1i 615 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | 
| 4 | r19.26 3110 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
| 5 | 3, 4 | sylibr 234 | 1 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wral 3060 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ral 3061 | 
| This theorem is referenced by: rr19.28v 3667 fununi 6640 txlm 23657 2reu8i 47130 2reuimp0 47131 | 
| Copyright terms: Public domain | W3C validator |