| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > r19.28zv | Structured version Visualization version GIF version | ||
| Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.) |
| Ref | Expression |
|---|---|
| r19.28zv | ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfv 1914 | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | 1 | r19.28z 4498 | 1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ≠ wne 2940 ∀wral 3061 ∅c0 4333 |
| 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-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-ne 2941 df-ral 3062 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: raaanv 4518 raltpd 4781 iinrab 5069 iindif2 5077 iinin2 5078 reusv2lem5 5402 xpiindi 5846 dfpo2 6316 fint 6787 ixpiin 8964 neips 23121 txflf 24014 isclmp 25130 diaglbN 41057 dihglbcpreN 41302 2reuimp 47127 |
| Copyright terms: Public domain | W3C validator |