Theorem dvelimor 1891

Description: Disjunctive distinct variable constraint elimination. A user of this theorem starts with a formula φ (containing z) and a distinct variable constraint between x and z. The theorem makes it possible to replace the distinct variable constraint with the disjunct ∀xx = y (ψ is just a version of φ with y substituted for z). (Contributed by Jim Kingdon, 11-May-2018.)

Hypotheses

Ref	Expression
dvelimor.1	⊢ Ⅎxφ
dvelimor.2	⊢ (z = y → (φ ↔ ψ))

Assertion

Ref	Expression
dvelimor	⊢ (∀x x = y ∨ Ⅎxψ)

Distinct variable groups:   ψ,z   x,z

Allowed substitution hints:   φ(x,y,z)   ψ(x,y)

Proof of Theorem dvelimor

Step	Hyp	Ref	Expression
1		ax-bndl 1396	. . . . . 6 ⊢ (∀x x = z ∨ (∀x x = y ∨ ∀z∀x(z = y → ∀x z = y)))
2		orcom 646	. . . . . . 7 ⊢ ((∀x x = y ∨ ∀z∀x(z = y → ∀x z = y)) ↔ (∀z∀x(z = y → ∀x z = y) ∨ ∀x x = y))
3	2	orbi2i 678	. . . . . 6 ⊢ ((∀x x = z ∨ (∀x x = y ∨ ∀z∀x(z = y → ∀x z = y))) ↔ (∀x x = z ∨ (∀z∀x(z = y → ∀x z = y) ∨ ∀x x = y)))
4	1, 3	mpbi 133	. . . . 5 ⊢ (∀x x = z ∨ (∀z∀x(z = y → ∀x z = y) ∨ ∀x x = y))
5		orass 683	. . . . 5 ⊢ (((∀x x = z ∨ ∀z∀x(z = y → ∀x z = y)) ∨ ∀x x = y) ↔ (∀x x = z ∨ (∀z∀x(z = y → ∀x z = y) ∨ ∀x x = y)))
6	4, 5	mpbir 134	. . . 4 ⊢ ((∀x x = z ∨ ∀z∀x(z = y → ∀x z = y)) ∨ ∀x x = y)
7		nfae 1604	. . . . . . 7 ⊢ Ⅎz∀x x = z
8		a16nf 1743	. . . . . . 7 ⊢ (∀x x = z → Ⅎx(z = y → φ))
9	7, 8	alrimi 1412	. . . . . 6 ⊢ (∀x x = z → ∀zℲx(z = y → φ))
10		df-nf 1347	. . . . . . . 8 ⊢ (Ⅎx z = y ↔ ∀x(z = y → ∀x z = y))
11		id 19	. . . . . . . . 9 ⊢ (Ⅎx z = y → Ⅎx z = y)
12		dvelimor.1	. . . . . . . . . 10 ⊢ Ⅎxφ
13	12	a1i 9	. . . . . . . . 9 ⊢ (Ⅎx z = y → Ⅎxφ)
14	11, 13	nfimd 1474	. . . . . . . 8 ⊢ (Ⅎx z = y → Ⅎx(z = y → φ))
15	10, 14	sylbir 125	. . . . . . 7 ⊢ (∀x(z = y → ∀x z = y) → Ⅎx(z = y → φ))
16	15	alimi 1341	. . . . . 6 ⊢ (∀z∀x(z = y → ∀x z = y) → ∀zℲx(z = y → φ))
17	9, 16	jaoi 635	. . . . 5 ⊢ ((∀x x = z ∨ ∀z∀x(z = y → ∀x z = y)) → ∀zℲx(z = y → φ))
18	17	orim1i 676	. . . 4 ⊢ (((∀x x = z ∨ ∀z∀x(z = y → ∀x z = y)) ∨ ∀x x = y) → (∀zℲx(z = y → φ) ∨ ∀x x = y))
19	6, 18	ax-mp 7	. . 3 ⊢ (∀zℲx(z = y → φ) ∨ ∀x x = y)
20		orcom 646	. . 3 ⊢ ((∀zℲx(z = y → φ) ∨ ∀x x = y) ↔ (∀x x = y ∨ ∀zℲx(z = y → φ)))
21	19, 20	mpbi 133	. 2 ⊢ (∀x x = y ∨ ∀zℲx(z = y → φ))
22		nfalt 1467	. . . 4 ⊢ (∀zℲx(z = y → φ) → Ⅎx∀z(z = y → φ))
23		ax-17 1416	. . . . . 6 ⊢ (ψ → ∀zψ)
24		dvelimor.2	. . . . . 6 ⊢ (z = y → (φ ↔ ψ))
25	23, 24	equsalh 1611	. . . . 5 ⊢ (∀z(z = y → φ) ↔ ψ)
26	25	nfbii 1359	. . . 4 ⊢ (Ⅎx∀z(z = y → φ) ↔ Ⅎxψ)
27	22, 26	sylib 127	. . 3 ⊢ (∀zℲx(z = y → φ) → Ⅎxψ)
28	27	orim2i 677	. 2 ⊢ ((∀x x = y ∨ ∀zℲx(z = y → φ)) → (∀x x = y ∨ Ⅎxψ))
29	21, 28	ax-mp 7	1 ⊢ (∀x x = y ∨ Ⅎxψ)

Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 628  ∀wal 1240  Ⅎwnf 1346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by:  nfsb4or  1896  rgen2a  2369