Theorem 19.24 1662

Description: Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem 19.24

Step	Hyp	Ref	Expression
1		19.2 1659	. . 3 ⊢ (∀xψ → ∃xψ)
2	1	imim2i 13	. 2 ⊢ ((∀xφ → ∀xψ) → (∀xφ → ∃xψ))
3		19.35 1600	. 2 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ∃xψ))
4	2, 3	sylibr 203	1 ⊢ ((∀xφ → ∀xψ) → ∃x(φ → ψ))

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-9 1654

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)