Theorem rexanali 2661

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

Assertion

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

Proof of Theorem rexanali

Step	Hyp	Ref	Expression
1		annim 414	. . 3 ⊢ ((φ ∧ ¬ ψ) ↔ ¬ (φ → ψ))
2	1	rexbii 2640	. 2 ⊢ (∃x ∈ A (φ ∧ ¬ ψ) ↔ ∃x ∈ A ¬ (φ → ψ))
3		rexnal 2626	. 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 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:  transex  5911  antisymex  5913  foundex  5915  extex  5916  symex  5917  nclennlem1  6249  nchoicelem16  6305