Theorem 19.20 992

Description: Theorem 19.20 of [Margaris] p. 90. (The proof was shortened by O'Cat, 30-Mar-2008.)

Assertion

Ref	Expression
19.20	⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))

Proof of Theorem 19.20

Step	Hyp	Ref	Expression
1		id 59	. . . 4 ⊢ ((φ → ψ) → (φ → ψ))
2	1	a4sd 983	. . 3 ⊢ ((φ → ψ) → (∀xφ → ψ))
3	2	19.20i 990	. 2 ⊢ (∀x(φ → ψ) → ∀x(∀xφ → ψ))
4		ax-5o 973	. 2 ⊢ (∀x(∀xφ → ψ) → (∀xφ → ∀xψ))
5	3, 4	syl 10	1 ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))

Syntax hints:   → wi 3  ∀wal 952

This theorem is referenced by:  19.20ii 993  19.21 1054  19.29 1069  19.30 1083  19.21t 1113  sbal1 1344  mo 1391  2mo 1445

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

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