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Mirrors > Home > ILE Home > Th. List > spimed | GIF version |
Description: Deduction version of spime 1720. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.) |
Ref | Expression |
---|---|
spimed.1 | ⊢ (𝜒 → Ⅎ𝑥𝜑) |
spimed.2 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
spimed | ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spimed.1 | . . 3 ⊢ (𝜒 → Ⅎ𝑥𝜑) | |
2 | 1 | nfrd 1501 | . 2 ⊢ (𝜒 → (𝜑 → ∀𝑥𝜑)) |
3 | a9e 1675 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
4 | spimed.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓)) | |
5 | 3, 4 | eximii 1582 | . . 3 ⊢ ∃𝑥(𝜑 → 𝜓) |
6 | 5 | 19.35i 1605 | . 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓) |
7 | 2, 6 | syl6 33 | 1 ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1330 Ⅎwnf 1437 ∃wex 1469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-i9 1511 ax-ial 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1438 |
This theorem is referenced by: spime 1720 |
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