Theorem rexnalim 2494

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 2489	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
2		exanaliim 1669	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3		df-ral 2488	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4	2, 3	sylnibr 678	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥 ∈ 𝐴 𝜑)
5	1, 4	sylbi 121	1 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1370  ∃wex 1514   ∈ wcel 2175  ∀wral 2483  ∃wrex 2484

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-ral 2488  df-rex 2489

This theorem is referenced by:  nnral  2495  ralexim  2497  iundif2ss  3992  ixp0  6808  omniwomnimkv  7251  alzdvds  12084  pc2dvds  12572  isnsgrp  13156  nninfsellemeq  15815