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Mirrors > Home > ILE Home > Th. List > spimth | GIF version |
Description: Closed theorem form of spim 1666. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
spimth | ⊢ (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imim2 54 | . . . . . 6 ⊢ ((𝜓 → ∀𝑥𝜓) → ((𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓))) | |
2 | 1 | imim2d 53 | . . . . 5 ⊢ ((𝜓 → ∀𝑥𝜓) → ((𝑥 = 𝑦 → (𝜑 → 𝜓)) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜓)))) |
3 | 2 | imp 122 | . . . 4 ⊢ (((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜓))) |
4 | 3 | com23 77 | . . 3 ⊢ (((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜓))) |
5 | 4 | al2imi 1387 | . 2 ⊢ (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓))) |
6 | ax9o 1628 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓) → 𝜓) | |
7 | 5, 6 | syl6 33 | 1 ⊢ (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1282 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-i9 1463 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: equveli 1682 |
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