Theorem ralexim 2361

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 2360	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∀𝑥 ∈ 𝐴 𝜑)
2	1	con2i 590	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wral 2349  ∃wrex 2350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-ral 2354  df-rex 2355

This theorem is referenced by: (None)