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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 1915 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.28z 4401 | 1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ≠ wne 2987 ∀wral 3106 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-ral 3111 df-dif 3884 df-nul 4244 |
This theorem is referenced by: raaanv 4419 raltpd 4677 iinrab 4954 iindif2 4962 iinin2 4963 reusv2lem5 5268 xpiindi 5670 fint 6532 ixpiin 8471 neips 21718 txflf 22611 isclmp 23702 dfpo2 33104 diaglbN 38351 dihglbcpreN 38596 2reuimp 43671 |
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