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Mirrors > Home > ILE Home > Th. List > nrex | GIF version |
Description: Inference adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.) |
Ref | Expression |
---|---|
nrex.1 | ⊢ (𝑥 ∈ 𝐴 → ¬ 𝜓) |
Ref | Expression |
---|---|
nrex | ⊢ ¬ ∃𝑥 ∈ 𝐴 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrex.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝜓) | |
2 | 1 | rgen 2547 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ¬ 𝜓 |
3 | ralnex 2482 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓) | |
4 | 2, 3 | mpbi 145 | 1 ⊢ ¬ ∃𝑥 ∈ 𝐴 𝜓 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2164 ∀wral 2472 ∃wrex 2473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-5 1458 ax-gen 1460 ax-ie2 1505 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-ral 2477 df-rex 2478 |
This theorem is referenced by: rex0 3464 iun0 3969 canth 5863 frec0g 6441 nominpos 9210 sqrt2irr 12290 gsum0g 12969 exmidsbthrlem 15457 |
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