Theorem rexalim 2389

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

Assertion

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

Proof of Theorem rexalim

Step	Hyp	Ref	Expression
1		ralnex 2385	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
2	1	biimpi 119	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝜑)
3	2	con2i 597	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wral 2375  ∃wrex 2376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-5 1391  ax-gen 1393  ax-ie2 1438

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-ral 2380  df-rex 2381

This theorem is referenced by:  infnlbti  6828