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Mirrors > Home > ILE Home > Th. List > 19.36-1 | GIF version |
Description: Closed form of 19.36i 1683. 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 1635 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) | |
2 | 19.36-1.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
3 | 2 | 19.9 1655 | . 2 ⊢ (∃𝑥𝜓 ↔ 𝜓) |
4 | 1, 3 | imbitrdi 161 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1362 Ⅎwnf 1471 ∃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 df-nf 1472 |
This theorem is referenced by: vtocl2 2807 vtocl3 2808 spcimgft 2828 |
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