Theorem 19.35 1600

Description: Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)

Assertion

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

Proof of Theorem 19.35

Step	Hyp	Ref	Expression
1		19.26 1593	. . . 4 ⊢ (∀x(φ ∧ ¬ ψ) ↔ (∀xφ ∧ ∀x ¬ ψ))
2		annim 414	. . . . 5 ⊢ ((φ ∧ ¬ ψ) ↔ ¬ (φ → ψ))
3	2	albii 1566	. . . 4 ⊢ (∀x(φ ∧ ¬ ψ) ↔ ∀x ¬ (φ → ψ))
4		alnex 1543	. . . . 5 ⊢ (∀x ¬ ψ ↔ ¬ ∃xψ)
5	4	anbi2i 675	. . . 4 ⊢ ((∀xφ ∧ ∀x ¬ ψ) ↔ (∀xφ ∧ ¬ ∃xψ))
6	1, 3, 5	3bitr3i 266	. . 3 ⊢ (∀x ¬ (φ → ψ) ↔ (∀xφ ∧ ¬ ∃xψ))
7		alnex 1543	. . 3 ⊢ (∀x ¬ (φ → ψ) ↔ ¬ ∃x(φ → ψ))
8		annim 414	. . 3 ⊢ ((∀xφ ∧ ¬ ∃xψ) ↔ ¬ (∀xφ → ∃xψ))
9	6, 7, 8	3bitr3i 266	. 2 ⊢ (¬ ∃x(φ → ψ) ↔ ¬ (∀xφ → ∃xψ))
10	9	con4bii 288	1 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ∃xψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀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:  19.35i  1601  19.35ri  1602  19.25  1603  19.43  1605  speimfw  1645  19.39  1661  19.24  1662  19.36  1871  19.37  1873  sbequi  2059