Theorem 19.21t 1113

Description: Closed form of Theorem 19.21 of [Margaris] p. 90.

Assertion

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

Proof of Theorem 19.21t

Step	Hyp	Ref	Expression
1		19.20 992	. . . . 5 ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))
2	1	imim2d 25	. . . 4 ⊢ (∀x(φ → ψ) → ((φ → ∀xφ) → (φ → ∀xψ)))
3	2	com12 11	. . 3 ⊢ ((φ → ∀xφ) → (∀x(φ → ψ) → (φ → ∀xψ)))
4	3	a4s 982	. 2 ⊢ (∀x(φ → ∀xφ) → (∀x(φ → ψ) → (φ → ∀xψ)))
5		hba1 1001	. . . 4 ⊢ (∀x(φ → ∀xφ) → ∀x∀x(φ → ∀xφ))
6		ax-4 971	. . . 4 ⊢ (∀x(φ → ∀xφ) → (φ → ∀xφ))
7		hba1 1001	. . . . 5 ⊢ (∀xψ → ∀x∀xψ)
8	7	a1i 8	. . . 4 ⊢ (∀x(φ → ∀xφ) → (∀xψ → ∀x∀xψ))
9	5, 6, 8	hbimd 1108	. . 3 ⊢ (∀x(φ → ∀xφ) → ((φ → ∀xψ) → ∀x(φ → ∀xψ)))
10		ax-4 971	. . . . 5 ⊢ (∀xψ → ψ)
11	10	imim2i 17	. . . 4 ⊢ ((φ → ∀xψ) → (φ → ψ))
12	11	19.20i 990	. . 3 ⊢ (∀x(φ → ∀xψ) → ∀x(φ → ψ))
13	9, 12	syl6 22	. 2 ⊢ (∀x(φ → ∀xφ) → ((φ → ∀xψ) → ∀x(φ → ψ)))
14	4, 13	impbid 515	1 ⊢ (∀x(φ → ∀xφ) → (∀x(φ → ψ) ↔ (φ → ∀xψ)))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 952

This theorem is referenced by:  sbcom 1256  sbal2 1356  ax11indalem 1366  ax11inda2ALT 1367  r19.21t 1712  sbciegft 1955

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-4 971  ax-5o 973  ax-6o 976

This theorem depends on definitions:  df-bi 147  df-an 225

Copyright terms: Public domain