Theorem 19.21 738

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "x is not free in φ".

Hypothesis

Ref	Expression
19.21.1	⊢ (φ → ∀xφ)

Assertion

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

Proof of Theorem 19.21

Step	Hyp	Ref	Expression
1		19.20 690	. . 3 ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))
2		19.21.1	. . 3 ⊢ (φ → ∀xφ)
3	1, 2	syl5 22	. 2 ⊢ (∀x(φ → ψ) → (φ → ∀xψ))
4		hba1 698	. . . 4 ⊢ (∀xψ → ∀x∀xψ)
5	2, 4	hbim 702	. . 3 ⊢ ((φ → ∀xψ) → ∀x(φ → ∀xψ))
6		ax-4 673	. . . . 5 ⊢ (∀xψ → ψ)
7	6	syl3 18	. . . 4 ⊢ ((φ → ∀xψ) → (φ → ψ))
8	7	19.20i 691	. . 3 ⊢ (∀x(φ → ∀xψ) → ∀x(φ → ψ))
9	5, 8	syl 12	. 2 ⊢ ((φ → ∀xψ) → ∀x(φ → ψ))
10	3, 9	impbi 139	1 ⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672

This theorem is referenced by:  stdpc5 739  19.32 765  hbim1 781  19.21v 942  cbvald 977  ax15 1006  eu2 1023  moanim 1051

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128

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