Theorem wl-mo2df 37941

Description: Version of mof 2567 with a context and a distinctor replacing a distinct variable condition. This version should be used only to eliminate disjoint variable conditions. (Contributed by Wolf Lammen, 11-Aug-2019.)

Hypotheses

Ref	Expression
wl-mo2df.1	⊢ Ⅎ𝑥𝜑
wl-mo2df.2	⊢ Ⅎ𝑦𝜑
wl-mo2df.3	⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)
wl-mo2df.4	⊢ (𝜑 → Ⅎ𝑦𝜓)

Assertion

Ref	Expression
wl-mo2df	⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)))

Proof of Theorem wl-mo2df

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfmo 2544	. 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑢∀𝑥(𝜓 → 𝑥 = 𝑢))
2		wl-mo2df.2	. . 3 ⊢ Ⅎ𝑦𝜑
3		wl-mo2df.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		wl-mo2df.4	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓)
5		wl-mo2df.3	. . . . . 6 ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)
6		nfeqf1 2387	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 = 𝑢)
7	6	naecoms 2437	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑢)
8	5, 7	syl 17	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦 𝑥 = 𝑢)
9	4, 8	nfimd 1901	. . . 4 ⊢ (𝜑 → Ⅎ𝑦(𝜓 → 𝑥 = 𝑢))
10	3, 9	nfald 2337	. . 3 ⊢ (𝜑 → Ⅎ𝑦∀𝑥(𝜓 → 𝑥 = 𝑢))
11		nfnae 2442	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
12		nfeqf2 2385	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑢 = 𝑦)
13	11, 12	nfan1 2212	. . . . . 6 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑢 = 𝑦)
14		equequ2 2033	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → (𝑥 = 𝑢 ↔ 𝑥 = 𝑦))
15	14	imbi2d 341	. . . . . . 7 ⊢ (𝑢 = 𝑦 → ((𝜓 → 𝑥 = 𝑢) ↔ (𝜓 → 𝑥 = 𝑦)))
16	15	adantl 482	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑢 = 𝑦) → ((𝜓 → 𝑥 = 𝑢) ↔ (𝜓 → 𝑥 = 𝑦)))
17	13, 16	albid 2234	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑢 = 𝑦) → (∀𝑥(𝜓 → 𝑥 = 𝑢) ↔ ∀𝑥(𝜓 → 𝑥 = 𝑦)))
18	5, 17	sylan 586	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑦) → (∀𝑥(𝜓 → 𝑥 = 𝑢) ↔ ∀𝑥(𝜓 → 𝑥 = 𝑦)))
19	18	ex 413	. . 3 ⊢ (𝜑 → (𝑢 = 𝑦 → (∀𝑥(𝜓 → 𝑥 = 𝑢) ↔ ∀𝑥(𝜓 → 𝑥 = 𝑦))))
20	2, 10, 19	cbvexd 2416	. 2 ⊢ (𝜑 → (∃𝑢∀𝑥(𝜓 → 𝑥 = 𝑢) ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)))
21	1, 20	bitrid 284	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545  ∃wex 1786  Ⅎwnf 1790  ∃*wmo 2541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-mo 2543

This theorem is referenced by:  wl-mo2tf  37942