Theorem rgen2a 2422

Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 needn't be distinct (and illustrates the use of dvelimor 1937). (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.)

Hypothesis

Ref	Expression
rgen2a.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑)

Assertion

Ref	Expression
rgen2a	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem rgen2a

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1462	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
2		eleq1 2145	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
3	1, 2	dvelimor 1937	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥 ∈ 𝐴)
4		eleq1 2145	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
5		rgen2a.1	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑)
6	5	ex 113	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝜑))
7	4, 6	syl6bi 161	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝜑)))
8	7	pm2.43d 49	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜑))
9	8	alimi 1385	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
10	9	a1d 22	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)))
11		nfr 1452	. . . . . 6 ⊢ (Ⅎ𝑦 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴))
12	6	alimi 1385	. . . . . 6 ⊢ (∀𝑦 𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
13	11, 12	syl6 33	. . . . 5 ⊢ (Ⅎ𝑦 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)))
14	10, 13	jaoi 669	. . . 4 ⊢ ((∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)))
15	3, 14	ax-mp 7	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
16		df-ral 2358	. . 3 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
17	15, 16	sylibr 132	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑)
18	17	rgen 2421	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑

Syntax hints:   → wi 4   ∧ wa 102   ∨ wo 662  ∀wal 1283   = wceq 1285  Ⅎwnf 1390   ∈ wcel 1434  ∀wral 2353

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-cleq 2076  df-clel 2079  df-ral 2358

This theorem is referenced by:  ordsucunielexmid  4302  onintexmid  4343  isoid  5501  issmo  5957  oawordriexmid  6134  ecopover  6291  ecopoverg  6294  1domsn  6384  unfiexmid  6462  subf  7429  negiso  8152  cnref1o  8866  ioof  9122  fzof  9283  gcdf  10571  eucalgf  10644  qredeu  10686