Theorem 19.37 1873

Description: Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
19.37.1	⊢ Ⅎxφ

Assertion

Ref	Expression
19.37	⊢ (∃x(φ → ψ) ↔ (φ → ∃xψ))

Proof of Theorem 19.37

Step	Hyp	Ref	Expression
1		19.35 1600	. 2 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ∃xψ))
2		19.37.1	. . . 4 ⊢ Ⅎxφ
3	2	19.3 1785	. . 3 ⊢ (∀xφ ↔ φ)
4	3	imbi1i 315	. 2 ⊢ ((∀xφ → ∃xψ) ↔ (φ → ∃xψ))
5	1, 4	bitri 240	1 ⊢ (∃x(φ → ψ) ↔ (φ → ∃xψ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is used by:  19.37v  1899