Theorem rgen2a 2604

Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2077. Usage of rgen2 2636 instead is highly encouraged. (Contributed by NM, 23-Nov-1994.) (Proof rewritten by Jim Kingdon, 1-Jun-2018.) (New usage is discouraged.)

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 1581	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
2		eleq1 2301	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
3	1, 2	dvelimor 2078	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥 ∈ 𝐴)
4		eleq1 2301	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
5		rgen2a.1	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑)
6	5	ex 115	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝜑))
7	4, 6	biimtrdi 163	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝜑)))
8	7	pm2.43d 50	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜑))
9	8	alimi 1508	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
10	9	a1d 22	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)))
11		nfr 1571	. . . . . 6 ⊢ (Ⅎ𝑦 𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴))
12	6	alimi 1508	. . . . . 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 728	. . . 4 ⊢ ((∀𝑦 𝑦 = 𝑥 ∨ Ⅎ𝑦 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)))
15	3, 14	ax-mp 5	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
16		df-ral 2533	. . 3 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
17	15, 16	sylibr 134	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑)
18	17	rgen 2603	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∨ wo 720  ∀wal 1400   = wceq 1402  Ⅎwnf 1513   ∈ wcel 2209  ∀wral 2528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-ral 2533

This theorem is used by:  ordsucunielexmid  4678  onintexmid  4720  isoid  6016  issmo  6559  oawordriexmid  6743  ecopover  6907  ecopoverg  6910  1domsn  7115  unfiexmid  7225  axaddf  8236  axmulf  8237  subf  8530  negiso  9288  cnref1o  10062  xaddf  10257  ioof  10384  fzof  10562  xrnegiso  12047  reeff1  12486  gcdf  12768  eucalgf  12852  qredeu  12894  qnnen  13374  strsetsid  13437  hmeofn  15494  ismeti  15538  qtopbasss  15713  tgqioo  15747  peano4nninf  17215