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Mirrors > Home > ILE Home > Th. List > 19.36-1 | GIF version |
Description: Closed form of 19.36i 1618. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.) |
Ref | Expression |
---|---|
19.36-1.1 | ⊢ Ⅎ𝑥𝜓 |
Ref | Expression |
---|---|
19.36-1 | ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.35-1 1571 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.36-1.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | 19.9 1591 | . 2 ⊢ (∃𝑥𝜓 ↔ 𝜓) |
4 | 1, 3 | syl6ib 160 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1297 Ⅎwnf 1404 ∃wex 1436 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-4 1455 ax-ial 1482 |
This theorem depends on definitions: df-bi 116 df-nf 1405 |
This theorem is referenced by: vtocl2 2696 vtocl3 2697 spcimgft 2717 |
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