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Theorem spimth 1670
Description: Closed theorem form of spim 1673. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)
Assertion
Ref Expression
spimth (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))

Proof of Theorem spimth
StepHypRef Expression
1 imim2 54 . . . . . 6 ((𝜓 → ∀𝑥𝜓) → ((𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
21imim2d 53 . . . . 5 ((𝜓 → ∀𝑥𝜓) → ((𝑥 = 𝑦 → (𝜑𝜓)) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜓))))
32imp 122 . . . 4 (((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑𝜓))) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜓)))
43com23 77 . . 3 (((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑𝜓))) → (𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜓)))
54al2imi 1392 . 2 (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓)))
6 ax9o 1633 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓) → 𝜓)
75, 6syl6 33 1 (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1287
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  equveli  1689
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