Theorem ralexim 2312

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 2311	. 2 ⊢ (∃x ∈ A ¬ φ → ¬ ∀x ∈ A φ)
2	1	con2i 557	1 ⊢ (∀x ∈ A φ → ¬ ∃x ∈ A ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4  ∀wral 2300  ∃wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-ral 2305  df-rex 2306

This theorem is referenced by: (None)