Theorem rexalim 2480

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 2475	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑)
2	1	biimpi 120	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∃𝑥 ∈ 𝐴 𝜑)
3	2	con2i 628	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wral 2465  ∃wrex 2466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1457  ax-gen 1459  ax-ie2 1504

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-ral 2470  df-rex 2471

This theorem is referenced by:  infnlbti  7038