Theorem 19.23 745

Description: Theorem 19.23 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.23.1	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
19.23	⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ))

Proof of Theorem 19.23

Step	Hyp	Ref	Expression
1		19.22 722	. . 3 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ))
2		19.23.1	. . . 4 ⊢ (ψ → ∀xψ)
3	2	19.9r 718	. . 3 ⊢ (ψ ↔ ∃xψ)
4	1, 3	syl6ibr 186	. 2 ⊢ (∀x(φ → ψ) → (∃xφ → ψ))
5		hbe1 709	. . . 4 ⊢ (∃xφ → ∀x∃xφ)
6	5, 2	hbim 702	. . 3 ⊢ ((∃xφ → ψ) → ∀x(∃xφ → ψ))
7		19.8a 712	. . . 4 ⊢ (φ → ∃xφ)
8	7	syl4 19	. . 3 ⊢ ((∃xφ → ψ) → (φ → ψ))
9	6, 8	19.21ai 740	. 2 ⊢ ((∃xφ → ψ) → ∀x(φ → ψ))
10	4, 9	impbi 139	1 ⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678

This theorem is referenced by:  19.23ad 748  sbied 903  19.23v 950  ceqsalg 1362  ralidm 1774

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