Theorem rgen2a 3360

Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2483. This theorem relies on the full set of axioms up to ax-ext 2735 and it should no longer be used. Usage of rgen2 3205 is highly encouraged. (Contributed by NM, 23-Nov-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 1-Jan-2020.) (Proof modification is discouraged.) (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		eleq1 2851	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
2	1	dvelimv 2484	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴))
3		rgen2a.1	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑)
4	3	ex 417	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝜑))
5	4	alimi 1841	. . . . 5 ⊢ (∀𝑦 𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
6	2, 5	syl6com 38	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)))
7		eleq1 2851	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
8	7	biimpd 232	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝐴))
9	8, 4	syli 40	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜑))
10	9	alimi 1841	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
11	6, 10	pm2.61d2 183	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
12		df-ral 3080	. . 3 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
13	11, 12	sylibr 237	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑)
14	13	rgen 3081	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400  ∀wal 1568   ∈ wcel 2143  ∀wral 3079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-13 2404  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-cleq 2755  df-clel 2838  df-ral 3080

This theorem is referenced by: (None)