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Mirrors > Home > ILE Home > Th. List > exlimd2 | GIF version |
Description: Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1576 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.) |
Ref | Expression |
---|---|
exlimd2.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
exlimd2.2 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
exlimd2.3 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
exlimd2 | ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exlimd2.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | exlimd2.2 | . . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
3 | 1, 2 | alrimih 1449 | . 2 ⊢ (𝜑 → ∀𝑥(𝜒 → ∀𝑥𝜒)) |
4 | exlimd2.3 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
5 | 1, 4 | alrimih 1449 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒)) |
6 | 19.23ht 1477 | . . 3 ⊢ (∀𝑥(𝜒 → ∀𝑥𝜒) → (∀𝑥(𝜓 → 𝜒) ↔ (∃𝑥𝜓 → 𝜒))) | |
7 | 6 | biimpd 143 | . 2 ⊢ (∀𝑥(𝜒 → ∀𝑥𝜒) → (∀𝑥(𝜓 → 𝜒) → (∃𝑥𝜓 → 𝜒))) |
8 | 3, 5, 7 | sylc 62 | 1 ⊢ (𝜑 → (∃𝑥𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1333 ∃wex 1472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-5 1427 ax-gen 1429 ax-ie2 1474 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: equsexd 1709 cbvexdh 1906 |
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