Theorem spimehOLD 1821

Description: Obsolete proof of spimeh 1667 as of 10-Dec-2017. (Contributed by NM, 7-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
spimehOLD.1	⊢ (φ → ∀xφ)
spimehOLD.2	⊢ (x = z → (φ → ψ))

Assertion

Ref	Expression
spimehOLD	⊢ (φ → ∃xψ)

Distinct variable group:   x,z

Allowed substitution hints:   φ(x,z)   ψ(x,z)

Proof of Theorem spimehOLD

Step	Hyp	Ref	Expression
1		ax9v 1655	. . . 4 ⊢ ¬ ∀x ¬ x = z
2		id 19	. . . . . . 7 ⊢ (φ → φ)
3	2	hbth 1552	. . . . . . . 8 ⊢ ((φ → φ) → ∀x(φ → φ))
4		hba1 1786	. . . . . . . . 9 ⊢ (∀x ¬ ψ → ∀x∀x ¬ ψ)
5	4	a1i 10	. . . . . . . 8 ⊢ ((φ → φ) → (∀x ¬ ψ → ∀x∀x ¬ ψ))
6		spimehOLD.1	. . . . . . . . . 10 ⊢ (φ → ∀xφ)
7	6	hbn 1776	. . . . . . . . 9 ⊢ (¬ φ → ∀x ¬ φ)
8	7	a1i 10	. . . . . . . 8 ⊢ ((φ → φ) → (¬ φ → ∀x ¬ φ))
9	3, 5, 8	hbimd 1815	. . . . . . 7 ⊢ ((φ → φ) → ((∀x ¬ ψ → ¬ φ) → ∀x(∀x ¬ ψ → ¬ φ)))
10	2, 9	ax-mp 5	. . . . . 6 ⊢ ((∀x ¬ ψ → ¬ φ) → ∀x(∀x ¬ ψ → ¬ φ))
11	10	hbn 1776	. . . . 5 ⊢ (¬ (∀x ¬ ψ → ¬ φ) → ∀x ¬ (∀x ¬ ψ → ¬ φ))
12		spimehOLD.2	. . . . . . 7 ⊢ (x = z → (φ → ψ))
13		sp 1747	. . . . . . 7 ⊢ (∀x ¬ ψ → ¬ ψ)
14	12, 13	nsyli 133	. . . . . 6 ⊢ (x = z → (∀x ¬ ψ → ¬ φ))
15	14	con3i 127	. . . . 5 ⊢ (¬ (∀x ¬ ψ → ¬ φ) → ¬ x = z)
16	11, 15	alrimih 1565	. . . 4 ⊢ (¬ (∀x ¬ ψ → ¬ φ) → ∀x ¬ x = z)
17	1, 16	mt3 171	. . 3 ⊢ (∀x ¬ ψ → ¬ φ)
18	17	con2i 112	. 2 ⊢ (φ → ¬ ∀x ¬ ψ)
19		df-ex 1542	. 2 ⊢ (∃xψ ↔ ¬ ∀x ¬ ψ)
20	18, 19	sylibr 203	1 ⊢ (φ → ∃xψ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541

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-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)