Theorem equs5or 1802

Description: Lemma used in proofs of substitution properties. Like equs5 1801 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 1674	. 2 ⊢ ∃𝑧 𝑧 = 𝑦
2		dveeq2or 1788	. . . . . 6 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ Ⅎ𝑥 𝑧 = 𝑦)
3		nfnf1 1523	. . . . . . . . . . 11 ⊢ Ⅎ𝑥Ⅎ𝑥 𝑧 = 𝑦
4	3	nfri 1499	. . . . . . . . . 10 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → ∀𝑥Ⅎ𝑥 𝑧 = 𝑦)
5		ax11v 1799	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)))
6		equequ2 1689	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
7	6	adantl 275	. . . . . . . . . . . . . 14 ⊢ ((Ⅎ𝑥 𝑧 = 𝑦 ∧ 𝑧 = 𝑦) → (𝑥 = 𝑧 ↔ 𝑥 = 𝑦))
8		nfr 1498	. . . . . . . . . . . . . . . . 17 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦))
9	8	imp 123	. . . . . . . . . . . . . . . 16 ⊢ ((Ⅎ𝑥 𝑧 = 𝑦 ∧ 𝑧 = 𝑦) → ∀𝑥 𝑧 = 𝑦)
10		hba1 1520	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 𝑧 = 𝑦 → ∀𝑥∀𝑥 𝑧 = 𝑦)
11	6	imbi1d 230	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)))
12	11	sps 1517	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 𝑧 = 𝑦 → ((𝑥 = 𝑧 → 𝜑) ↔ (𝑥 = 𝑦 → 𝜑)))
13	10, 12	albidh 1456	. . . . . . . . . . . . . . . 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 229	. . . . . . . . . . . . . 14 ⊢ ((Ⅎ𝑥 𝑧 = 𝑦 ∧ 𝑧 = 𝑦) → ((𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
16	7, 15	imbi12d 233	. . . . . . . . . . . . 13 ⊢ ((Ⅎ𝑥 𝑧 = 𝑦 ∧ 𝑧 = 𝑦) → ((𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) ↔ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
17	5, 16	mpbii 147	. . . . . . . . . . . 12 ⊢ ((Ⅎ𝑥 𝑧 = 𝑦 ∧ 𝑧 = 𝑦) → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
18	17	ex 114	. . . . . . . . . . 11 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))))
19	18	imp4a 346	. . . . . . . . . 10 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
20	4, 19	alrimih 1445	. . . . . . . . 9 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → ∀𝑥(𝑧 = 𝑦 → ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
21		19.21t 1561	. . . . . . . . 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 146	. . . . . . . 8 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
23		hba1 1520	. . . . . . . . 9 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑦 → 𝜑))
24	23	19.23h 1474	. . . . . . . 8 ⊢ (∀𝑥((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
25	22, 24	syl6ib 160	. . . . . . 7 ⊢ (Ⅎ𝑥 𝑧 = 𝑦 → (𝑧 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
26	25	orim2i 750	. . . . . 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 5	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ (𝑧 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
28		pm2.76 797	. . . . 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 5	. . . 4 ⊢ ((∀𝑥 𝑥 = 𝑦 ∨ 𝑧 = 𝑦) → (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
30	29	olcs 725	. . 3 ⊢ (𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
31	30	exlimiv 1577	. 2 ⊢ (∃𝑧 𝑧 = 𝑦 → (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))))
32	1, 31	ax-mp 5	1 ⊢ (∀𝑥 𝑥 = 𝑦 ∨ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  ∀wal 1329  Ⅎwnf 1436  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  sb4or  1805