Theorem nnral 2460

Description: The double negation of a universal quantification implies the universal quantification of the double negation. Restricted quantifier version of nnal 1642. (Contributed by Jim Kingdon, 1-Aug-2024.)

Assertion

Ref	Expression
nnral	⊢ (¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)

Proof of Theorem nnral

Step	Hyp	Ref	Expression
1		rexnalim 2459	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∀𝑥 ∈ 𝐴 𝜑)
2	1	con3i 627	. 2 ⊢ (¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
3		ralnex 2458	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
4	2, 3	sylibr 133	1 ⊢ (¬ ¬ ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ¬ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wral 2448  ∃wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-ral 2453  df-rex 2454

This theorem is referenced by:  onntri13  7202  onntri24  7206