Theorem equs5or 1693

Description: Lemma used in proofs of substitution properties. Like equs5 1692 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.)

Assertion

Ref	Expression
equs5or	⊢ (∀x x = y ∨ (∃x(x = y ∧ φ) → ∀x(x = y → φ)))

Proof of Theorem equs5or

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		a9e 1568	. 2 ⊢ ∃z z = y
2		dveeq2or 1679	. . . . . 6 ⊢ (∀x x = y ∨ Ⅎx z = y)
3		nfnf1 1418	. . . . . . . . . . 11 ⊢ ℲxℲx z = y
4	3	nfri 1393	. . . . . . . . . 10 ⊢ (Ⅎx z = y → ∀xℲx z = y)
5		ax11v 1690	. . . . . . . . . . . . 13 ⊢ (x = z → (φ → ∀x(x = z → φ)))
6		equequ2 1581	. . . . . . . . . . . . . . 15 ⊢ (z = y → (x = z ↔ x = y))
7	6	adantl 262	. . . . . . . . . . . . . 14 ⊢ ((Ⅎx z = y ∧ z = y) → (x = z ↔ x = y))
8		nfr 1392	. . . . . . . . . . . . . . . . 17 ⊢ (Ⅎx z = y → (z = y → ∀x z = y))
9	8	imp 115	. . . . . . . . . . . . . . . 16 ⊢ ((Ⅎx z = y ∧ z = y) → ∀x z = y)
10		hba1 1415	. . . . . . . . . . . . . . . . 17 ⊢ (∀x z = y → ∀x∀x z = y)
11	6	imbi1d 220	. . . . . . . . . . . . . . . . . 18 ⊢ (z = y → ((x = z → φ) ↔ (x = y → φ)))
12	11	sps 1412	. . . . . . . . . . . . . . . . 17 ⊢ (∀x z = y → ((x = z → φ) ↔ (x = y → φ)))
13	10, 12	albidh 1349	. . . . . . . . . . . . . . . 16 ⊢ (∀x z = y → (∀x(x = z → φ) ↔ ∀x(x = y → φ)))
14	9, 13	syl 14	. . . . . . . . . . . . . . 15 ⊢ ((Ⅎx z = y ∧ z = y) → (∀x(x = z → φ) ↔ ∀x(x = y → φ)))
15	14	imbi2d 219	. . . . . . . . . . . . . 14 ⊢ ((Ⅎx z = y ∧ z = y) → ((φ → ∀x(x = z → φ)) ↔ (φ → ∀x(x = y → φ))))
16	7, 15	imbi12d 223	. . . . . . . . . . . . 13 ⊢ ((Ⅎx z = y ∧ z = y) → ((x = z → (φ → ∀x(x = z → φ))) ↔ (x = y → (φ → ∀x(x = y → φ)))))
17	5, 16	mpbii 136	. . . . . . . . . . . 12 ⊢ ((Ⅎx z = y ∧ z = y) → (x = y → (φ → ∀x(x = y → φ))))
18	17	ex 108	. . . . . . . . . . 11 ⊢ (Ⅎx z = y → (z = y → (x = y → (φ → ∀x(x = y → φ)))))
19	18	imp4a 331	. . . . . . . . . 10 ⊢ (Ⅎx z = y → (z = y → ((x = y ∧ φ) → ∀x(x = y → φ))))
20	4, 19	alrimih 1338	. . . . . . . . 9 ⊢ (Ⅎx z = y → ∀x(z = y → ((x = y ∧ φ) → ∀x(x = y → φ))))
21		19.21t 1456	. . . . . . . . 9 ⊢ (Ⅎx z = y → (∀x(z = y → ((x = y ∧ φ) → ∀x(x = y → φ))) ↔ (z = y → ∀x((x = y ∧ φ) → ∀x(x = y → φ)))))
22	20, 21	mpbid 135	. . . . . . . 8 ⊢ (Ⅎx z = y → (z = y → ∀x((x = y ∧ φ) → ∀x(x = y → φ))))
23		hba1 1415	. . . . . . . . 9 ⊢ (∀x(x = y → φ) → ∀x∀x(x = y → φ))
24	23	19.23h 1368	. . . . . . . 8 ⊢ (∀x((x = y ∧ φ) → ∀x(x = y → φ)) ↔ (∃x(x = y ∧ φ) → ∀x(x = y → φ)))
25	22, 24	syl6ib 150	. . . . . . 7 ⊢ (Ⅎx z = y → (z = y → (∃x(x = y ∧ φ) → ∀x(x = y → φ))))
26	25	orim2i 665	. . . . . 6 ⊢ ((∀x x = y ∨ Ⅎx z = y) → (∀x x = y ∨ (z = y → (∃x(x = y ∧ φ) → ∀x(x = y → φ)))))
27	2, 26	ax-mp 7	. . . . 5 ⊢ (∀x x = y ∨ (z = y → (∃x(x = y ∧ φ) → ∀x(x = y → φ))))
28		pm2.76 708	. . . . 5 ⊢ ((∀x x = y ∨ (z = y → (∃x(x = y ∧ φ) → ∀x(x = y → φ)))) → ((∀x x = y ∨ z = y) → (∀x x = y ∨ (∃x(x = y ∧ φ) → ∀x(x = y → φ)))))
29	27, 28	ax-mp 7	. . . 4 ⊢ ((∀x x = y ∨ z = y) → (∀x x = y ∨ (∃x(x = y ∧ φ) → ∀x(x = y → φ))))
30	29	olcs 642	. . 3 ⊢ (z = y → (∀x x = y ∨ (∃x(x = y ∧ φ) → ∀x(x = y → φ))))
31	30	exlimiv 1471	. 2 ⊢ (∃z z = y → (∀x x = y ∨ (∃x(x = y ∧ φ) → ∀x(x = y → φ))))
32	1, 31	ax-mp 7	1 ⊢ (∀x x = y ∨ (∃x(x = y ∧ φ) → ∀x(x = y → φ)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616  ∀wal 1226  Ⅎwnf 1329  ∃wex 1362

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628

This theorem is referenced by:  sb4or  1696