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Mirrors > Home > ILE Home > Th. List > spimt | GIF version |
Description: Closed theorem form of spim 1626. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.) |
Ref | Expression |
---|---|
spimt | ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | a9e 1586 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exim 1490 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑 → 𝜓))) | |
3 | 1, 2 | mpi 15 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ∃𝑥(𝜑 → 𝜓)) |
4 | 19.35-1 1515 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
6 | 19.9t 1533 | . . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓)) | |
7 | 6 | biimpd 132 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓)) |
8 | 5, 7 | sylan9r 390 | 1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 = wceq 1243 Ⅎwnf 1349 ∃wex 1381 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-i9 1423 ax-ial 1427 |
This theorem depends on definitions: df-bi 110 df-nf 1350 |
This theorem is referenced by: spimd 9905 |
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