Theorem ralinexa 2980

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 437	. . 3 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2	1	ralbii 2963	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓))
3		ralnex 2975	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
4	2, 3	bitri 263	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wral 2896  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ral 2901  df-rex 2902

This theorem is referenced by:  kmlem7  8861  kmlem13  8867  lspsncv0  18967  ntreq0  20691  lhop1lem  23580  soseq  30995  ltrnnid  34440