Theorem rexnalim 2311

Description: Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)

Assertion

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

Proof of Theorem rexnalim

Step	Hyp	Ref	Expression
1		df-rex 2306	. 2 ⊢ (∃x ∈ A ¬ φ ↔ ∃x(x ∈ A ∧ ¬ φ))
2		exanaliim 1535	. . 3 ⊢ (∃x(x ∈ A ∧ ¬ φ) → ¬ ∀x(x ∈ A → φ))
3		df-ral 2305	. . 3 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
4	2, 3	sylnibr 601	. 2 ⊢ (∃x(x ∈ A ∧ ¬ φ) → ¬ ∀x ∈ A φ)
5	1, 4	sylbi 114	1 ⊢ (∃x ∈ A ¬ φ → ¬ ∀x ∈ A φ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀wal 1240  ∃wex 1378   ∈ wcel 1390  ∀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:  ralexim  2312  iundif2ss  3713