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Mirrors > Home > ILE Home > Th. List > r19.28v | GIF version |
Description: Restricted quantifier version of one direction of 19.28 1525. (The other direction holds when 𝐴 is inhabited, see r19.28mv 3421.) (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 19 | . . . 4 ⊢ (𝜑 → 𝜑) | |
2 | 1 | ralrimivw 2480 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜑) |
3 | 2 | anim1i 336 | . 2 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
4 | r19.26 2532 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
5 | 3, 4 | sylibr 133 | 1 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wral 2390 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1406 ax-gen 1408 ax-4 1470 ax-17 1489 |
This theorem depends on definitions: df-bi 116 df-nf 1420 df-ral 2395 |
This theorem is referenced by: txlm 12290 |
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