Theorem 19.21hOLD 1840

Description: Obsolete proof of 19.21h 1797 as of 1-Jan-2018. (Contributed by NM, 1-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.21hOLD.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
19.21hOLD	⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ))

Proof of Theorem 19.21hOLD

Step	Hyp	Ref	Expression
1		19.21hOLD.1	. . 3 ⊢ (φ → ∀xφ)
2		alim 1558	. . 3 ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))
3	1, 2	syl5 28	. 2 ⊢ (∀x(φ → ψ) → (φ → ∀xψ))
4		hba1 1786	. . . 4 ⊢ (∀xψ → ∀x∀xψ)
5	1, 4	hbim 1817	. . 3 ⊢ ((φ → ∀xψ) → ∀x(φ → ∀xψ))
6		sp 1747	. . . 4 ⊢ (∀xψ → ψ)
7	6	imim2i 13	. . 3 ⊢ ((φ → ∀xψ) → (φ → ψ))
8	5, 7	alrimih 1565	. 2 ⊢ ((φ → ∀xψ) → ∀x(φ → ψ))
9	3, 8	impbii 180	1 ⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540

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)