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Mirrors > Home > ILE Home > Th. List > 19.37-1 | GIF version |
Description: One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.) |
Ref | Expression |
---|---|
19.37-1.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.37-1 | ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.37-1.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | 19.3 1564 | . 2 ⊢ (∀𝑥𝜑 ↔ 𝜑) |
3 | 19.35-1 1634 | . 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) | |
4 | 2, 3 | biimtrrid 153 | 1 ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1361 Ⅎwnf 1470 ∃wex 1502 |
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 1457 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-4 1520 ax-ial 1544 |
This theorem depends on definitions: df-bi 117 df-nf 1471 |
This theorem is referenced by: 19.37aiv 1685 spcimegft 2827 eqvincg 2873 |
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