Theorem 19.35 754

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.

Assertion

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

Proof of Theorem 19.35

Step	Hyp	Ref	Expression
1		19.26 749	. . . 4 ⊢ (∀x(φ ∧ ¬ ψ) ↔ (∀xφ ∧ ∀x ¬ ψ))
2		annim 206	. . . . 5 ⊢ ((φ ∧ ¬ ψ) ↔ ¬ (φ → ψ))
3	2	bial 695	. . . 4 ⊢ (∀x(φ ∧ ¬ ψ) ↔ ∀x ¬ (φ → ψ))
4		df-an 198	. . . 4 ⊢ ((∀xφ ∧ ∀x ¬ ψ) ↔ ¬ (∀xφ → ¬ ∀x ¬ ψ))
5	1, 3, 4	3bitr3 156	. . 3 ⊢ (∀x ¬ (φ → ψ) ↔ ¬ (∀xφ → ¬ ∀x ¬ ψ))
6	5	bicon2i 194	. 2 ⊢ ((∀xφ → ¬ ∀x ¬ ψ) ↔ ¬ ∀x ¬ (φ → ψ))
7		df-ex 679	. . 3 ⊢ (∃xψ ↔ ¬ ∀x ¬ ψ)
8	7	imbi2i 160	. 2 ⊢ ((∀xφ → ∃xψ) ↔ (∀xφ → ¬ ∀x ¬ ψ))
9		df-ex 679	. 2 ⊢ (∃x(φ → ψ) ↔ ¬ ∀x ¬ (φ → ψ))
10	6, 8, 9	3bitr4r 159	1 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ∃xψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678

This theorem is referenced by:  19.35i 755  19.35ri 756  19.36 757  19.37 759  19.39 761  19.24 762  19.25 763  sbequi 876

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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