Theorem 19.20 690

Description: Theorem 19.20 of [Margaris] p. 90.

Assertion

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

Proof of Theorem 19.20

Step	Hyp	Ref	Expression
1		ax-4 673	. . . . 5 ⊢ (∀xφ → φ)
2	1	syl4 19	. . . 4 ⊢ ((φ → ψ) → (∀xφ → ψ))
3	2	a4s 682	. . 3 ⊢ (∀x(φ → ψ) → (∀xφ → ψ))
4	3	a5i 687	. 2 ⊢ (∀x(φ → ψ) → ∀x(∀xφ → ψ))
5		ax-5 674	. 2 ⊢ (∀x(∀xφ → ψ) → (∀xφ → ∀xψ))
6	4, 5	syl 12	1 ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))

Syntax hints:   → wi 2  ∀wal 672

This theorem is referenced by:  19.20ii 692  19.21 738  19.29 752  19.30 764  19.21g 792  sbal1 996  mo 1020

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

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