Theorem wl-eudf 37072

Description: Version of eu6 2563 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, 23-Sep-2020.)

Hypotheses

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

Assertion

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

Proof of Theorem wl-eudf

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eu6 2563	. 2 ⊢ (∃!𝑥𝜓 ↔ ∃𝑢∀𝑥(𝜓 ↔ 𝑥 = 𝑢))
2		wl-eudf.2	. . 3 ⊢ Ⅎ𝑦𝜑
3		wl-eudf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
4		wl-eudf.4	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦𝜓)
5		wl-eudf.3	. . . . . 6 ⊢ (𝜑 → ¬ ∀𝑥 𝑥 = 𝑦)
6		nfeqf1 2373	. . . . . . 7 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑥 = 𝑢)
7	6	naecoms 2423	. . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑢)
8	5, 7	syl 17	. . . . 5 ⊢ (𝜑 → Ⅎ𝑦 𝑥 = 𝑢)
9	4, 8	nfbid 1897	. . . 4 ⊢ (𝜑 → Ⅎ𝑦(𝜓 ↔ 𝑥 = 𝑢))
10	3, 9	nfald 2316	. . 3 ⊢ (𝜑 → Ⅎ𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑢))
11		nfnae 2428	. . . . . . 7 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
12		nfeqf2 2371	. . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑢 = 𝑦)
13	11, 12	nfan1 2188	. . . . . 6 ⊢ Ⅎ𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑢 = 𝑦)
14		equequ2 2021	. . . . . . . 8 ⊢ (𝑢 = 𝑦 → (𝑥 = 𝑢 ↔ 𝑥 = 𝑦))
15	14	bibi2d 341	. . . . . . 7 ⊢ (𝑢 = 𝑦 → ((𝜓 ↔ 𝑥 = 𝑢) ↔ (𝜓 ↔ 𝑥 = 𝑦)))
16	15	adantl 480	. . . . . 6 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑢 = 𝑦) → ((𝜓 ↔ 𝑥 = 𝑢) ↔ (𝜓 ↔ 𝑥 = 𝑦)))
17	13, 16	albid 2210	. . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ 𝑢 = 𝑦) → (∀𝑥(𝜓 ↔ 𝑥 = 𝑢) ↔ ∀𝑥(𝜓 ↔ 𝑥 = 𝑦)))
18	5, 17	sylan 578	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑦) → (∀𝑥(𝜓 ↔ 𝑥 = 𝑢) ↔ ∀𝑥(𝜓 ↔ 𝑥 = 𝑦)))
19	18	ex 411	. . 3 ⊢ (𝜑 → (𝑢 = 𝑦 → (∀𝑥(𝜓 ↔ 𝑥 = 𝑢) ↔ ∀𝑥(𝜓 ↔ 𝑥 = 𝑦))))
20	2, 10, 19	cbvexd 2402	. 2 ⊢ (𝜑 → (∃𝑢∀𝑥(𝜓 ↔ 𝑥 = 𝑢) ↔ ∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦)))
21	1, 20	bitrid 282	1 ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 ↔ 𝑥 = 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394  ∀wal 1531  ∃wex 1773  Ⅎwnf 1777  ∃!weu 2557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2366

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-mo 2529  df-eu 2558

This theorem is referenced by:  wl-eutf  37073