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Mirrors > Home > ILE Home > Th. List > nrexdv | GIF version |
Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.) |
Ref | Expression |
---|---|
nrexdv.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝜓) |
Ref | Expression |
---|---|
nrexdv | ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrexdv.1 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝜓) | |
2 | 1 | ralrimiva 2505 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝜓) |
3 | ralnex 2426 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓) | |
4 | 2, 3 | sylib 121 | 1 ⊢ (𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-5 1423 ax-gen 1425 ax-ie2 1470 ax-4 1487 ax-17 1506 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-ral 2421 df-rex 2422 |
This theorem is referenced by: ltpopr 7403 cauappcvgprlemladdru 7464 cauappcvgprlemladdrl 7465 caucvgprlemladdrl 7486 caucvgprprlemaddq 7516 dvdsle 11542 |
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