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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 2212. (The other direction holds when 𝐴 is nonempty, see r19.27zv 4293.) (Contributed by NM, 3-Jun-2004.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
r19.27v | ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1 6 | . . . 4 ⊢ (𝜓 → (𝑥 ∈ 𝐴 → 𝜓)) | |
2 | 1 | ralrimiv 3146 | . . 3 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 𝜓) |
3 | 2 | anim2i 610 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) |
4 | r19.26 3249 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)) | |
5 | 3, 4 | sylibr 226 | 1 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ 𝜓) → ∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2106 ∀wral 3089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ral 3094 |
This theorem is referenced by: r19.28v 3256 txlm 21860 tx1stc 21862 spanuni 28975 |
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