Theorem r19.35 1298

Description: Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90.

Assertion

Ref	Expression
r19.35	⊢ (∃x ∈ A (φ → ψ) ↔ (∀x ∈ A φ → ∃x ∈ A ψ))

Proof of Theorem r19.35

Step	Hyp	Ref	Expression
1		r19.26 1289	. . . 4 ⊢ (∀x ∈ A (φ ∧ ¬ ψ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ¬ ψ))
2		annim 206	. . . . 5 ⊢ ((φ ∧ ¬ ψ) ↔ ¬ (φ → ψ))
3	2	biral 1223	. . . 4 ⊢ (∀x ∈ A (φ ∧ ¬ ψ) ↔ ∀x ∈ A ¬ (φ → ψ))
4		df-an 198	. . . 4 ⊢ ((∀x ∈ A φ ∧ ∀x ∈ A ¬ ψ) ↔ ¬ (∀x ∈ A φ → ¬ ∀x ∈ A ¬ ψ))
5	1, 3, 4	3bitr3 156	. . 3 ⊢ (∀x ∈ A ¬ (φ → ψ) ↔ ¬ (∀x ∈ A φ → ¬ ∀x ∈ A ¬ ψ))
6	5	bicon2i 194	. 2 ⊢ ((∀x ∈ A φ → ¬ ∀x ∈ A ¬ ψ) ↔ ¬ ∀x ∈ A ¬ (φ → ψ))
7		dfrex2 1212	. . 3 ⊢ (∃x ∈ A ψ ↔ ¬ ∀x ∈ A ¬ ψ)
8	7	imbi2i 160	. 2 ⊢ ((∀x ∈ A φ → ∃x ∈ A ψ) ↔ (∀x ∈ A φ → ¬ ∀x ∈ A ¬ ψ))
9		dfrex2 1212	. 2 ⊢ (∃x ∈ A (φ → ψ) ↔ ¬ ∀x ∈ A ¬ (φ → ψ))
10	6, 8, 9	3bitr4r 159	1 ⊢ (∃x ∈ A (φ → ψ) ↔ (∀x ∈ A φ → ∃x ∈ A ψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wral 1201  ∃wrex 1202

This theorem is referenced by:  r19.36av 1299  r19.37av 1300  r19.36zv 1772

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-or 197  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

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