Theorem eunex 4545

Description: Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by Jim Kingdon, 29-Dec-2018.)

Assertion

Ref	Expression
eunex	⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Proof of Theorem eunex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1521	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	eu3 2065	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
3		dtruex 4543	. . . . 5 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
4		nfa1 1534	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝑦)
5		sp 1504	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
6	5	con3d 626	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (¬ 𝑥 = 𝑦 → ¬ 𝜑))
7	4, 6	eximd 1605	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥 ¬ 𝑥 = 𝑦 → ∃𝑥 ¬ 𝜑))
8	3, 7	mpi 15	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
9	8	exlimiv 1591	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
10	9	adantl 275	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → ∃𝑥 ¬ 𝜑)
11	2, 10	sylbi 120	1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1346   = wceq 1348  ∃wex 1485  ∃!weu 2019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589

This theorem is referenced by: (None)