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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 2548 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) |
4 | rexim 2571 | . 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 1460 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 |
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-5 1447 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-4 1510 ax-ial 1534 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-ral 2460 df-rex 2461 |
This theorem is referenced by: reximdvai 2577 bezoutlemstep 11981 isomninnlem 14434 ismkvnnlem 14456 |
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