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 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))

Proof of Theorem spimth

Step	Hyp	Ref	Expression
1		imim2 49	. . . . . 6 ⊢ ((𝜓 → ∀𝑥𝜓) → ((𝜑 → 𝜓) → (𝜑 → ∀𝑥𝜓)))
2	1	imim2d 48	. . . . 5 ⊢ ((𝜓 → ∀𝑥𝜓) → ((𝑥 = 𝑦 → (𝜑 → 𝜓)) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜓))))
3	2	imp 115	. . . 4 ⊢ (((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜓)))
4	3	com23 72	. . 3 ⊢ (((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (𝜑 → (𝑥 = 𝑦 → ∀𝑥𝜓)))
5	4	al2imi 1347	. 2 ⊢ (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓)))
6		ax9o 1588	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜓) → 𝜓)
7	5, 6	syl6 29	1 ⊢ (∀𝑥((𝜓 → ∀𝑥𝜓) ∧ (𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))

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