Theorem spimed 1977

Description: Deduction version of spime 1976. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
spimed.1	⊢ (χ → Ⅎxφ)
spimed.2	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
spimed	⊢ (χ → (φ → ∃xψ))

Proof of Theorem spimed

Step	Hyp	Ref	Expression
1		spimed.1	. 2 ⊢ (χ → Ⅎxφ)
2		nfnf1 1790	. . . . 5 ⊢ ℲxℲxφ
3		id 19	. . . . 5 ⊢ (Ⅎxφ → Ⅎxφ)
4	2, 3	nfan1 1881	. . . 4 ⊢ Ⅎx(Ⅎxφ ∧ φ)
5		spimed.2	. . . . 5 ⊢ (x = y → (φ → ψ))
6	5	adantld 453	. . . 4 ⊢ (x = y → ((Ⅎxφ ∧ φ) → ψ))
7	4, 6	spime 1976	. . 3 ⊢ ((Ⅎxφ ∧ φ) → ∃xψ)
8	7	ex 423	. 2 ⊢ (Ⅎxφ → (φ → ∃xψ))
9	1, 8	syl 15	1 ⊢ (χ → (φ → ∃xψ))

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541  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:  equvini  1987