Theorem eunex 4377

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 1466	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	eu3 1994	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
3		dtruex 4375	. . . . 5 ⊢ ∃𝑥 ¬ 𝑥 = 𝑦
4		nfa1 1479	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝑦)
5		sp 1446	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
6	5	con3d 596	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (¬ 𝑥 = 𝑦 → ¬ 𝜑))
7	4, 6	eximd 1548	. . . . 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 1534	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑥 ¬ 𝜑)
10	9	adantl 271	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)) → ∃𝑥 ¬ 𝜑)
11	2, 10	sylbi 119	1 ⊢ (∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102  ∀wal 1287   = wceq 1289  ∃wex 1426  ∃!weu 1948

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-setind 4353

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452

This theorem is referenced by: (None)