Theorem mo2 2507

Description: Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.) Restrict dummy variable z. (Revised by Wolf Lammen, 28-May-2019.)

Hypothesis

Ref	Expression
mo2.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
mo2	⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mo2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo2v 2505	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))
2		mo2.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
3		nfv 1883	. . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧
4	2, 3	nfim 1865	. . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧)
5	4	nfal 2191	. . 3 ⊢ Ⅎ𝑦∀𝑥(𝜑 → 𝑥 = 𝑧)
6		nfv 1883	. . 3 ⊢ Ⅎ𝑧∀𝑥(𝜑 → 𝑥 = 𝑦)
7		equequ2 1999	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
8	7	imbi2d 329	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜑 → 𝑥 = 𝑦)))
9	8	albidv 1889	. . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦)))
10	5, 6, 9	cbvex 2308	. 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
11	1, 10	bitri 264	1 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521  ∃wex 1744  Ⅎwnf 1748  ∃*wmo 2499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503

This theorem is referenced by:  mo3  2536  mo  2537  rmo2  3559  nmo  29453  bj-eu3f  32954  bj-mo3OLD  32957  dffun3f  42754