Theorem rexnalim 2486

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 2481	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
2		exanaliim 1661	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3		df-ral 2480	. . 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 1362  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-ral 2480  df-rex 2481

This theorem is referenced by:  nnral  2487  ralexim  2489  iundif2ss  3983  ixp0  6799  omniwomnimkv  7242  alzdvds  12036  pc2dvds  12524  isnsgrp  13108  nninfsellemeq  15745