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Mirrors > Home > MPE Home > Th. List > spimt | Structured version Visualization version GIF version |
Description: Closed theorem form of spim 2319. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Mar-2023.) |
Ref | Expression |
---|---|
spimt | ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6e 2314 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exim 1797 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑 → 𝜓))) | |
3 | 1, 2 | mpi 20 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ∃𝑥(𝜑 → 𝜓)) |
4 | 19.35 1841 | . . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | |
5 | 3, 4 | sylib 210 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓)) |
6 | id 22 | . . 3 ⊢ (Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓) | |
7 | 6 | 19.9d 2133 | . 2 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓)) |
8 | 5, 7 | sylan9r 501 | 1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∀wal 1506 ∃wex 1743 Ⅎwnf 1747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-12 2107 ax-13 2302 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1744 df-nf 1748 |
This theorem is referenced by: (None) |
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