Theorem wl-mo2t 38507

Description: Closed form of mof 2589. (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		dfmo 2566	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑢∀𝑥(𝜑 → 𝑥 = 𝑢))
2		nfnf1 2191	. . . 4 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑
3	2	nfal 2354	. . 3 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑
4		nfa1 2188	. . . 4 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑
5		sp 2220	. . . . 5 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦𝜑)
6		nfvd 1948	. . . . 5 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦 𝑥 = 𝑢)
7	5, 6	nfimd 1927	. . . 4 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦(𝜑 → 𝑥 = 𝑢))
8	4, 7	nfald 2359	. . 3 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦∀𝑥(𝜑 → 𝑥 = 𝑢))
9		equequ2 2059	. . . . . 6 ⊢ (𝑢 = 𝑦 → (𝑥 = 𝑢 ↔ 𝑥 = 𝑦))
10	9	imbi2d 343	. . . . 5 ⊢ (𝑢 = 𝑦 → ((𝜑 → 𝑥 = 𝑢) ↔ (𝜑 → 𝑥 = 𝑦)))
11	10	albidv 1953	. . . 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	cbvexdw 2369	. 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃𝑢∀𝑥(𝜑 → 𝑥 = 𝑢) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
14	1, 13	bitrid 286	1 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816  ∃*wmo 2563

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-11 2194  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-mo 2565

This theorem is used by:  wl-mo3t  38508