Theorem mof 2556

Description: Version of df-mo 2533 with disjoint variable condition replaced by nonfreeness hypothesis. (Contributed by NM, 8-Mar-1995.) Extract dfmo 2589 from this proof, and prove mof 2556 from it (as of 30-Sep-2022, directly from df-mo 2533). (Revised by Wolf Lammen, 28-May-2019.) Avoid ax-13 2370. (Revised by Wolf Lammen, 16-Oct-2022.)

Hypothesis

Ref	Expression
mof.1	⊢ Ⅎ𝑦𝜑

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem mof

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mo 2533	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))
2		mof.1	. . . . 5 ⊢ Ⅎ𝑦𝜑
3		nfv 1917	. . . . 5 ⊢ Ⅎ𝑦 𝑥 = 𝑧
4	2, 3	nfim 1899	. . . 4 ⊢ Ⅎ𝑦(𝜑 → 𝑥 = 𝑧)
5	4	nfal 2316	. . 3 ⊢ Ⅎ𝑦∀𝑥(𝜑 → 𝑥 = 𝑧)
6		nfv 1917	. . 3 ⊢ Ⅎ𝑧∀𝑥(𝜑 → 𝑥 = 𝑦)
7		equequ2 2029	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
8	7	imbi2d 340	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝑥 = 𝑧) ↔ (𝜑 → 𝑥 = 𝑦)))
9	8	albidv 1923	. . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦)))
10	5, 6, 9	cbvexv1 2338	. 2 ⊢ (∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
11	1, 10	bitri 274	1 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539  ∃wex 1781  Ⅎwnf 1785  ∃*wmo 2531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ex 1782  df-nf 1786  df-mo 2533

This theorem is referenced by:  mo3  2557  mo  2558  rmo2  3846  nmo  31482  fineqvrep  33785  bj-eu3f  35383  dffun3f  47247