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| Description: Generalization rule for restricted quantification. Note that 𝑥 and 𝑦 are not required to be disjoint. This proof illustrates the use of dvelim 2456. This theorem relies on the full set of axioms up to ax-ext 2708 and it should no longer be used. Usage of rgen2 3199 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.) | 
| Ref | Expression | 
|---|---|
| rgen2a.1 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝜑) | 
| Ref | Expression | 
|---|---|
| rgen2a | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eleq1 2829 | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
| 2 | 1 | dvelimv 2457 | . . . . 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 1811 | . . . . 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 2829 | . . . . . . 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 1811 | . . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) | 
| 11 | 6, 10 | pm2.61d2 181 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) | 
| 12 | df-ral 3062 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜑)) | |
| 13 | 11, 12 | sylibr 234 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐴 𝜑) | 
| 14 | 13 | rgen 3063 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 ∈ wcel 2108 ∀wral 3061 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-cleq 2729 df-clel 2816 df-ral 3062 | 
| This theorem is referenced by: (None) | 
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