Theorem spimt 1974

Description: Closed theorem form of spim 1975. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.)

Assertion

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

Proof of Theorem spimt

Step	Hyp	Ref	Expression
1		nfnf1 1790	. . . . 5 ⊢ ℲxℲxψ
2		nfa1 1788	. . . . 5 ⊢ Ⅎx∀xφ
3	1, 2	nfan 1824	. . . 4 ⊢ Ⅎx(Ⅎxψ ∧ ∀xφ)
4		sp 1747	. . . . . . 7 ⊢ (∀xφ → φ)
5	4	adantl 452	. . . . . 6 ⊢ ((Ⅎxψ ∧ ∀xφ) → φ)
6		nfr 1761	. . . . . . 7 ⊢ (Ⅎxψ → (ψ → ∀xψ))
7	6	adantr 451	. . . . . 6 ⊢ ((Ⅎxψ ∧ ∀xφ) → (ψ → ∀xψ))
8	5, 7	embantd 50	. . . . 5 ⊢ ((Ⅎxψ ∧ ∀xφ) → ((φ → ψ) → ∀xψ))
9	8	imim2d 48	. . . 4 ⊢ ((Ⅎxψ ∧ ∀xφ) → ((x = y → (φ → ψ)) → (x = y → ∀xψ)))
10	3, 9	alimd 1764	. . 3 ⊢ ((Ⅎxψ ∧ ∀xφ) → (∀x(x = y → (φ → ψ)) → ∀x(x = y → ∀xψ)))
11	10	impancom 427	. 2 ⊢ ((Ⅎxψ ∧ ∀x(x = y → (φ → ψ))) → (∀xφ → ∀x(x = y → ∀xψ)))
12		ax9o 1950	. 2 ⊢ (∀x(x = y → ∀xψ) → ψ)
13	11, 12	syl6 29	1 ⊢ ((Ⅎxψ ∧ ∀x(x = y → (φ → ψ))) → (∀xφ → ψ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  spim  1975  equveli  1988