Theorem wl-mo2t 33357

Description: Closed form of mo2 2479. (Contributed by Wolf Lammen, 18-Aug-2019.)

Assertion

Ref	Expression
wl-mo2t	⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-mo2t

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mo2v 2477	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑢∀𝑥(𝜑 → 𝑥 = 𝑢))
2		nfnf1 2031	. . . 4 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
3	2	nfal 2153	. . 3 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
4		nfa1 2028	. . . 4 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
5		sp 2053	. . . . 5 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
6		nfvd 1844	. . . . 5 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦 𝑥 = 𝑢)
7	5, 6	nfimd 1823	. . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦(𝜑 → 𝑥 = 𝑢))
8	4, 7	nfald 2165	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥(𝜑 → 𝑥 = 𝑢))
9		equequ2 1953	. . . . . 6 ⊢ (𝑢 = 𝑦 → (𝑥 = 𝑢 ↔ 𝑥 = 𝑦))
10	9	imbi2d 330	. . . . 5 ⊢ (𝑢 = 𝑦 → ((𝜑 → 𝑥 = 𝑢) ↔ (𝜑 → 𝑥 = 𝑦)))
11	10	albidv 1849	. . . 4 ⊢ (𝑢 = 𝑦 → (∀𝑥(𝜑 → 𝑥 = 𝑢) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦)))
12	11	a1i 11	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (𝑢 = 𝑦 → (∀𝑥(𝜑 → 𝑥 = 𝑢) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑦))))
13	3, 8, 12	cbvexd 2278	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑢∀𝑥(𝜑 → 𝑥 = 𝑢) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
14	1, 13	syl5bb 272	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by:  wl-mo3t  33358