Theorem spimt 2317

Description: Closed theorem form of spim 2319. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 21-Mar-2023.)

Assertion

Ref	Expression
spimt	⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))

Proof of Theorem spimt

Step	Hyp	Ref	Expression
1		ax6e 2314	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
2		exim 1797	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥(𝜑 → 𝜓)))
3	1, 2	mpi 20	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ∃𝑥(𝜑 → 𝜓))
4		19.35 1841	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
5	3, 4	sylib 210	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∃𝑥𝜓))
6		id 22	. . 3 ⊢ (Ⅎ𝑥𝜓 → Ⅎ𝑥𝜓)
7	6	19.9d 2133	. 2 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 → 𝜓))
8	5, 7	sylan9r 501	1 ⊢ ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 387  ∀wal 1506  ∃wex 1743  Ⅎwnf 1747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-12 2107  ax-13 2302

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1744  df-nf 1748

This theorem is referenced by: (None)