Theorem 19.38 1083

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 1018	. . 3 ⊢ (∃xφ → ∀x∃xφ)
2		hba1 1005	. . 3 ⊢ (∀xψ → ∀x∀xψ)
3	1, 2	hbim 1009	. 2 ⊢ ((∃xφ → ∀xψ) → ∀x(∃xφ → ∀xψ))
4		19.8a 1031	. . 3 ⊢ (φ → ∃xφ)
5		ax-4 975	. . 3 ⊢ (∀xψ → ψ)
6	4, 5	imim12i 18	. 2 ⊢ ((∃xφ → ∀xψ) → (φ → ψ))
7	3, 6	19.21ai 1000	1 ⊢ ((∃xφ → ∀xψ) → ∀x(φ → ψ))

Syntax hints:   → wi 3  ∀wal 956  ∃wex 982

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977  ax-6o 980

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983

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