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Mirrors > Home > MPE Home > Th. List > r19.27v | Structured version Visualization version GIF version |
Description: Restricted quantitifer version of one direction of 19.27 2219. (The other direction holds when 𝐴 is nonempty, see r19.27zv 4447.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.) (Proof shortened by Wolf Lammen, 17-Jun-2023.) |
Ref | Expression |
---|---|
r19.27v | ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (𝜓 → 𝜓) | |
2 | 1 | ralrimivw 3180 | . . 3 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜓) |
3 | 2 | anim2i 616 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
4 | r19.26 3167 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
5 | 3, 4 | sylibr 235 | 1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wral 3135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ral 3140 |
This theorem is referenced by: r19.28vOLD 3184 txlm 22184 tx1stc 22186 spanuni 29248 |
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