Theorem euf 2565

Description: Version of eu6 2563 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 30-Oct-2018.) Avoid ax-13 2366. (Revised by Wolf Lammen, 16-Oct-2022.)

Hypothesis

Ref	Expression
euf.1	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
euf	⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem euf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eu6 2563	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧))
2		euf.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
3		nfv 1910	. . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧
4	2, 3	nfbi 1899	. . . 4 ⊢ Ⅎ𝑦(𝜑 ↔ 𝑥 = 𝑧)
5	4	nfal 2311	. . 3 ⊢ Ⅎ𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑧)
6		nfv 1910	. . 3 ⊢ Ⅎ𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
7		equequ2 2022	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
8	7	bibi2d 342	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑧) ↔ (𝜑 ↔ 𝑥 = 𝑦)))
9	8	albidv 1916	. . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
10	5, 6, 9	cbvexv1 2333	. 2 ⊢ (∃𝑧∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
11	1, 10	bitri 275	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

Syntax hints:   ↔ wb 205  ∀wal 1532  ∃wex 1774  Ⅎwnf 1778  ∃!weu 2557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2164

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-mo 2529  df-eu 2558

This theorem is referenced by:  eu1  2601