Theorem 19.23tOLD 1819

Description: Obsolete proof of 19.23t 1800 as of 1-Jan-2018. (Contributed by NM, 7-Nov-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
19.23tOLD	⊢ (Ⅎxψ → (∀x(φ → ψ) ↔ (∃xφ → ψ)))

Proof of Theorem 19.23tOLD

Step	Hyp	Ref	Expression
1		exim 1575	. . 3 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ))
2		19.9t 1779	. . . 4 ⊢ (Ⅎxψ → (∃xψ ↔ ψ))
3	2	imbi2d 307	. . 3 ⊢ (Ⅎxψ → ((∃xφ → ∃xψ) ↔ (∃xφ → ψ)))
4	1, 3	syl5ib 210	. 2 ⊢ (Ⅎxψ → (∀x(φ → ψ) → (∃xφ → ψ)))
5		nfnf1 1790	. . 3 ⊢ ℲxℲxψ
6		nfe1 1732	. . . . 5 ⊢ Ⅎx∃xφ
7	6	a1i 10	. . . 4 ⊢ (Ⅎxψ → Ⅎx∃xφ)
8		id 19	. . . 4 ⊢ (Ⅎxψ → Ⅎxψ)
9	7, 8	nfimd 1808	. . 3 ⊢ (Ⅎxψ → Ⅎx(∃xφ → ψ))
10		19.8a 1756	. . . . 5 ⊢ (φ → ∃xφ)
11	10	a1i 10	. . . 4 ⊢ (Ⅎxψ → (φ → ∃xφ))
12	11	imim1d 69	. . 3 ⊢ (Ⅎxψ → ((∃xφ → ψ) → (φ → ψ)))
13	5, 9, 12	alrimdd 1768	. 2 ⊢ (Ⅎxψ → ((∃xφ → ψ) → ∀x(φ → ψ)))
14	4, 13	impbid 183	1 ⊢ (Ⅎxψ → (∀x(φ → ψ) ↔ (∃xφ → ψ)))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃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-11 1746

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

This theorem is referenced by: (None)