Theorem rgen2a 3337

Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2451. This theorem relies on the full set of axioms up to ax-ext 2703 and it should no longer be used. Usage of rgen2 3172 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 2819	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
2	1	dvelimv 2452	. . . . 5 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴))
3		rgen2a.1	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑)
4	3	ex 412	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐴 → 𝜑))
5	4	alimi 1812	. . . . 5 ⊢ (∀𝑦 𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
6	2, 5	syl6com 37	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)))
7		eleq1 2819	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
8	7	biimpd 229	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝐴))
9	8, 4	syli 39	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 → 𝜑))
10	9	alimi 1812	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
11	6, 10	pm2.61d2 181	. . 3 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
12		df-ral 3048	. . 3 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑))
13	11, 12	sylibr 234	. 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑)
14	13	rgen 3049	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1539   ∈ wcel 2111  ∀wral 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-13 2372  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-cleq 2723  df-clel 2806  df-ral 3048

This theorem is referenced by: (None)