Theorem rexnalim 2360

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 2355	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
2		exanaliim 1579	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3		df-ral 2354	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4	2, 3	sylnibr 635	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥 ∈ 𝐴 𝜑)
5	1, 4	sylbi 119	1 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102  ∀wal 1283  ∃wex 1422   ∈ wcel 1434  ∀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:  ralexim  2361  iundif2ss  3751  alzdvds  10399