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Mirrors > Home > ILE Home > Th. List > 19.35-1 | GIF version |
Description: Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic (Contributed by Mario Carneiro, 2-Feb-2015.) |
Ref | Expression |
---|---|
19.35-1 | ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.29 1597 | . . 3 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥(𝜑 → 𝜓)) → ∃𝑥(𝜑 ∧ (𝜑 → 𝜓))) | |
2 | pm3.35 345 | . . . 4 ⊢ ((𝜑 ∧ (𝜑 → 𝜓)) → 𝜓) | |
3 | 2 | eximi 1577 | . . 3 ⊢ (∃𝑥(𝜑 ∧ (𝜑 → 𝜓)) → ∃𝑥𝜓) |
4 | 1, 3 | syl 14 | . 2 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥(𝜑 → 𝜓)) → ∃𝑥𝜓) |
5 | 4 | expcom 115 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1330 ∃wex 1469 |
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-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1487 ax-ial 1511 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 19.35i 1602 19.25 1603 19.36-1 1650 19.37-1 1651 spimt 1713 sbequi 1816 |
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