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Mirrors > Home > ILE Home > Th. List > spimt | GIF version |
Description: Closed theorem form of spim 1717. (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 1675 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exim 1579 | . . . 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 1604 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
6 | 19.9t 1622 | . . 3 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓)) | |
7 | 6 | biimpd 143 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓)) |
8 | 5, 7 | sylan9r 408 | 1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1330 = wceq 1332 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: spimd 13143 |
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