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Mirrors > Home > ILE Home > Th. List > eximdh | GIF version |
Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.) |
Ref | Expression |
---|---|
eximdh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
eximdh.2 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
eximdh | ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eximdh.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | eximdh.2 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 1, 2 | alrimih 1398 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
4 | exim 1530 | . 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∃𝑥𝜓 → ∃𝑥𝜒)) | |
5 | 3, 4 | syl 14 | 1 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 ∃wex 1421 |
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 |
This theorem is referenced by: eximd 1543 19.41h 1615 hbexd 1624 equsex 1656 equsexd 1657 spimeh 1667 sbiedh 1710 exdistrfor 1721 eximdv 1801 cbvexdh 1842 mopick2 2024 2euex 2028 bj-sbimedh 10582 |
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