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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 1631 | . . 3 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥(𝜑 → 𝜓)) → ∃𝑥(𝜑 ∧ (𝜑 → 𝜓))) | |
2 | pm3.35 347 | . . . 4 ⊢ ((𝜑 ∧ (𝜑 → 𝜓)) → 𝜓) | |
3 | 2 | eximi 1611 | . . 3 ⊢ (∃𝑥(𝜑 ∧ (𝜑 → 𝜓)) → ∃𝑥𝜓) |
4 | 1, 3 | syl 14 | . 2 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥(𝜑 → 𝜓)) → ∃𝑥𝜓) |
5 | 4 | expcom 116 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1362 ∃wex 1503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: 19.35i 1636 19.25 1637 19.36-1 1684 19.37-1 1685 spimt 1747 sbequi 1850 |
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