Theorem ralinexa 3089

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 399	. . 3 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓))
2	1	ralbii 3082	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓))
3		ralnex 3062	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝜑 ∧ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
4	2, 3	bitri 275	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wral 3051  ∃wrex 3060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-ral 3052  df-rex 3061

This theorem is referenced by:  soseq  8101  kmlem7  10067  kmlem13  10073  lspsncv0  21101  ntreq0  23021  lhop1lem  25974  nogt01o  27664  ltrnnid  40396