Theorem ralexim 2522

Description: Relationship between restricted universal and existential quantifiers. (Contributed by Jim Kingdon, 17-Aug-2018.)

Assertion

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

Proof of Theorem ralexim

Step	Hyp	Ref	Expression
1		rexnalim 2519	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∀𝑥 ∈ 𝐴 𝜑)
2	1	con2i 630	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wral 2508  ∃wrex 2509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by: (None)