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Mirrors > Home > ILE Home > Th. List > i19.39 | GIF version |
Description: Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1611, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.) |
Ref | Expression |
---|---|
i19.24.1 | ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) |
Ref | Expression |
---|---|
i19.39 | ⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.2 1625 | . . 3 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑) | |
2 | 1 | imim1i 60 | . 2 ⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
3 | i19.24.1 | . 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) | |
4 | 2, 3 | syl 14 | 1 ⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1340 ∃wex 1479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: (None) |
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