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| Description: Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.) | 
| Ref | Expression | 
|---|---|
| dfint2 | ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-int 4946 | . 2 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} | |
| 2 | df-ral 3061 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)) | |
| 3 | 2 | abbii 2808 | . 2 ⊢ {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} = {𝑥 ∣ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ∈ 𝑦)} | 
| 4 | 1, 3 | eqtr4i 2767 | 1 ⊢ ∩ 𝐴 = {𝑥 ∣ ∀𝑦 ∈ 𝐴 𝑥 ∈ 𝑦} | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 = wceq 1539 ∈ wcel 2107 {cab 2713 ∀wral 3060 ∩ cint 4945 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-ral 3061 df-int 4946 | 
| This theorem is referenced by: inteq 4948 elintg 4953 nfint 4955 intss 4968 intiin 5058 dfint3 35954 | 
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