Theorem wl-mo2df 38034

Description: Version of mof 2589 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 2566	. 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 2409	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 = 𝑢)
7	6	naecoms 2459	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑢)
8	5, 7	syl 17	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦 𝑥 = 𝑢)
9	4, 8	nfimd 1913	. . . 4 ⊢ (𝜑 → Ⅎ𝑦(𝜓 → 𝑥 = 𝑢))
10	3, 9	nfald 2359	. . 3 ⊢ (𝜑 → Ⅎ𝑦∀𝑥(𝜓 → 𝑥 = 𝑢))
11		nfnae 2464	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
12		nfeqf2 2407	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑢 = 𝑦)
13	11, 12	nfan1 2234	. . . . . 6 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑢 = 𝑦)
14		equequ2 2045	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → (𝑥 = 𝑢 ↔ 𝑥 = 𝑦))
15	14	imbi2d 342	. . . . . . 7 ⊢ (𝑢 = 𝑦 → ((𝜓 → 𝑥 = 𝑢) ↔ (𝜓 → 𝑥 = 𝑦)))
16	15	adantl 485	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑢 = 𝑦) → ((𝜓 → 𝑥 = 𝑢) ↔ (𝜓 → 𝑥 = 𝑦)))
17	13, 16	albid 2256	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑢 = 𝑦) → (∀𝑥(𝜓 → 𝑥 = 𝑢) ↔ ∀𝑥(𝜓 → 𝑥 = 𝑦)))
18	5, 17	sylan 589	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑦) → (∀𝑥(𝜓 → 𝑥 = 𝑢) ↔ ∀𝑥(𝜓 → 𝑥 = 𝑦)))
19	18	ex 416	. . 3 ⊢ (𝜑 → (𝑢 = 𝑦 → (∀𝑥(𝜓 → 𝑥 = 𝑢) ↔ ∀𝑥(𝜓 → 𝑥 = 𝑦))))
20	2, 10, 19	cbvexd 2438	. 2 ⊢ (𝜑 → (∃𝑢∀𝑥(𝜓 → 𝑥 = 𝑢) ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)))
21	1, 20	bitrid 285	1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557  ∃wex 1798  Ⅎwnf 1802  ∃*wmo 2563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-mo 2565

This theorem is referenced by:  wl-mo2tf  38035