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Mirrors > Home > ILE Home > Th. List > exlimdh | GIF version |
Description: Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.) |
Ref | Expression |
---|---|
exlimdh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
exlimdh.2 | ⊢ (𝜒 → ∀𝑥𝜒) |
exlimdh.3 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
exlimdh | ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exlimdh.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | exlimdh.3 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 1, 2 | alrimih 1358 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
4 | exlimdh.2 | . . 3 ⊢ (𝜒 → ∀𝑥𝜒) | |
5 | 4 | 19.23h 1387 | . 2 ⊢ (∀𝑥(𝜓 → 𝜒) ↔ (∃𝑥𝜓 → 𝜒)) |
6 | 3, 5 | sylib 127 | 1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
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-5 1336 ax-gen 1338 ax-ie2 1383 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: exlimd 1488 exim 1490 exlimdv 1700 equs5 1710 cbvexdh 1801 exists2 1997 |
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