Theorem ralinexa 2659

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)

Assertion

Ref	Expression
ralinexa	⊢ (∀x ∈ A (φ → ¬ ψ) ↔ ¬ ∃x ∈ A (φ ∧ ψ))

Proof of Theorem ralinexa

Step	Hyp	Ref	Expression
1		imnan 411	. . 3 ⊢ ((φ → ¬ ψ) ↔ ¬ (φ ∧ ψ))
2	1	ralbii 2638	. 2 ⊢ (∀x ∈ A (φ → ¬ ψ) ↔ ∀x ∈ A ¬ (φ ∧ ψ))
3		ralnex 2624	. 2 ⊢ (∀x ∈ A ¬ (φ ∧ ψ) ↔ ¬ ∃x ∈ A (φ ∧ ψ))
4	2, 3	bitri 240	1 ⊢ (∀x ∈ A (φ → ¬ ψ) ↔ ¬ ∃x ∈ A (φ ∧ ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wral 2614  ∃wrex 2615

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 2619  df-rex 2620

This theorem is referenced by: (None)