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Mirrors > Home > ILE Home > Th. List > 19.36-1 | GIF version |
Description: Closed form of 19.36i 1603. 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 1556 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.36-1.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | 19.9 1576 | . 2 ⊢ (∃𝑥𝜓 ↔ 𝜓) |
4 | 1, 3 | syl6ib 159 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1283 Ⅎwnf 1390 ∃wex 1422 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1377 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-4 1441 ax-ial 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1391 |
This theorem is referenced by: vtocl2 2663 vtocl3 2664 spcimgft 2683 |
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