Theorem r19.35 2759

Description: Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 20-Sep-2003.)

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 2747	. . . 4 ⊢ (∀x ∈ A (φ ∧ ¬ ψ) ↔ (∀x ∈ A φ ∧ ∀x ∈ A ¬ ψ))
2		annim 414	. . . . 5 ⊢ ((φ ∧ ¬ ψ) ↔ ¬ (φ → ψ))
3	2	ralbii 2639	. . . 4 ⊢ (∀x ∈ A (φ ∧ ¬ ψ) ↔ ∀x ∈ A ¬ (φ → ψ))
4		df-an 360	. . . 4 ⊢ ((∀x ∈ A φ ∧ ∀x ∈ A ¬ ψ) ↔ ¬ (∀x ∈ A φ → ¬ ∀x ∈ A ¬ ψ))
5	1, 3, 4	3bitr3i 266	. . 3 ⊢ (∀x ∈ A ¬ (φ → ψ) ↔ ¬ (∀x ∈ A φ → ¬ ∀x ∈ A ¬ ψ))
6	5	con2bii 322	. 2 ⊢ ((∀x ∈ A φ → ¬ ∀x ∈ A ¬ ψ) ↔ ¬ ∀x ∈ A ¬ (φ → ψ))
7		dfrex2 2628	. . 3 ⊢ (∃x ∈ A ψ ↔ ¬ ∀x ∈ A ¬ ψ)
8	7	imbi2i 303	. 2 ⊢ ((∀x ∈ A φ → ∃x ∈ A ψ) ↔ (∀x ∈ A φ → ¬ ∀x ∈ A ¬ ψ))
9		dfrex2 2628	. 2 ⊢ (∃x ∈ A (φ → ψ) ↔ ¬ ∀x ∈ A ¬ (φ → ψ))
10	6, 8, 9	3bitr4ri 269	1 ⊢ (∃x ∈ A (φ → ψ) ↔ (∀x ∈ A φ → ∃x ∈ A ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wral 2615  ∃wrex 2616

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by:  r19.36av  2760  r19.37  2761  r19.43  2767  r19.37zv  3647  r19.36zv  3651