Theorem rexnalim 2519

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 2514	. 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
2		exanaliim 1693	. . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝜑) → ¬ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
3		df-ral 2513	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4	2, 3	sylnibr 681	. 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 1393  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509

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 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-ral 2513  df-rex 2514

This theorem is referenced by:  nnral  2520  ralexim  2522  rexanaliim  2636  iundif2ss  4030  ixp0  6876  omniwomnimkv  7330  alzdvds  12360  pc2dvds  12848  isnsgrp  13434  umgr2edg1  16001  umgr2edgneu  16004  nninfsellemeq  16339