Theorem spimth 1694

Description: Closed theorem form of spim 1697. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)

Assertion

Ref	Expression
spimth	⊢ (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))

Proof of Theorem spimth

Step	Hyp	Ref	Expression
1		imim2 55	. . . . . 6 ⊢ ((𝜓 → ∀𝑥𝜓) → ((𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
2	1	imim2d 54	. . . . 5 ⊢ ((𝜓 → ∀𝑥𝜓) → ((𝑥 = 𝑦 → (𝜑 → 𝜓)) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜓))))
3	2	imp 123	. . . 4 ⊢ (((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜓)))
4	3	com23 78	. . 3 ⊢ (((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜓)))
5	4	al2imi 1415	. 2 ⊢ (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓)))
6		ax9o 1657	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓) → 𝜓)
7	5, 6	syl6 33	1 ⊢ (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1310

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-4 1468  ax-i9 1491  ax-ial 1495

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  equveli  1713