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Theorem spimth 1623
 Description: Closed theorem form of spim 1626. (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 49 . . . . . 6 ((𝜓 → ∀𝑥𝜓) → ((𝜑𝜓) → (𝜑 → ∀𝑥𝜓)))
21imim2d 48 . . . . 5 ((𝜓 → ∀𝑥𝜓) → ((𝑥 = 𝑦 → (𝜑𝜓)) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜓))))
32imp 115 . . . 4 (((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑𝜓))) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜓)))
43com23 72 . . 3 (((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑𝜓))) → (𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜓)))
54al2imi 1347 . 2 (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓)))
6 ax9o 1588 . 2 (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓) → 𝜓)
75, 6syl6 29 1 (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑𝜓))) → (∀𝑥𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241 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 This theorem is referenced by:  equveli  1642
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