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Mirrors > Home > ILE Home > Th. List > reximdai | GIF version |
Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999.) |
Ref | Expression |
---|---|
reximdai.1 | ⊢ Ⅎ𝑥𝜑 |
reximdai.2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) |
Ref | Expression |
---|---|
reximdai | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reximdai.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | reximdai.2 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒))) | |
3 | 1, 2 | ralrimi 2432 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
4 | rexim 2455 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜓 → 𝜒) → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) | |
5 | 3, 4 | syl 14 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Ⅎwnf 1389 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-ral 2353 df-rex 2354 |
This theorem is referenced by: reximdvai 2461 bezoutlemstep 10386 |
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