Theorem 19.25 763

Description: Theorem 19.25 of [Margaris] p. 90.

Assertion

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

Proof of Theorem 19.25

Step	Hyp	Ref	Expression
1		19.35 754	. . . 4 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ∃xψ))
2	1	biimp 133	. . 3 ⊢ (∃x(φ → ψ) → (∀xφ → ∃xψ))
3	2	19.20i 691	. 2 ⊢ (∀y∃x(φ → ψ) → ∀y(∀xφ → ∃xψ))
4		19.22 722	. 2 ⊢ (∀y(∀xφ → ∃xψ) → (∃y∀xφ → ∃y∃xψ))
5	3, 4	syl 12	1 ⊢ (∀y∃x(φ → ψ) → (∃y∀xφ → ∃y∃xψ))

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-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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