Theorem 19.38 760

Description: Theorem 19.38 of [Margaris] p. 90.

Assertion

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

Proof of Theorem 19.38

Step	Hyp	Ref	Expression
1		hbe1 709	. . 3 ⊢ (∃xφ → ∀x∃xφ)
2		hba1 698	. . 3 ⊢ (∀xψ → ∀x∀xψ)
3	1, 2	hbim 702	. 2 ⊢ ((∃xφ → ∀xψ) → ∀x(∃xφ → ∀xψ))
4		19.8a 712	. . 3 ⊢ (φ → ∃xφ)
5		ax-4 673	. . 3 ⊢ (∀xψ → ψ)
6	4, 5	syl34 20	. 2 ⊢ ((∃xφ → ∀xψ) → (φ → ψ))
7	3, 6	19.21ai 740	1 ⊢ ((∃xφ → ∀xψ) → ∀x(φ → ψ))

Syntax hints:   → wi 2  ∀wal 672  ∃wex 678

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  df-an 198  df-ex 679

metamath.org