Theorem 19.25 1603

Description: Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem 19.25

Step	Hyp	Ref	Expression
1		19.35 1600	. . . 4 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ∃xψ))
2	1	biimpi 186	. . 3 ⊢ (∃x(φ → ψ) → (∀xφ → ∃xψ))
3	2	alimi 1559	. 2 ⊢ (∀y∃x(φ → ψ) → ∀y(∀xφ → ∃xψ))
4		exim 1575	. 2 ⊢ (∀y(∀xφ → ∃xψ) → (∃y∀xφ → ∃y∃xψ))
5	3, 4	syl 15	1 ⊢ (∀y∃x(φ → ψ) → (∃y∀xφ → ∃y∃xψ))

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

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

This theorem is referenced by: (None)