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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 1358 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
4 | exim 1490 | . 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 1241 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: eximd 1503 19.41h 1575 hbexd 1584 equsex 1616 equsexd 1617 spimeh 1627 sbiedh 1670 exdistrfor 1681 eximdv 1760 cbvexdh 1801 mopick2 1983 2euex 1987 bj-sbimedh 9911 |
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