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| Description: Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.) | 
| Ref | Expression | 
|---|---|
| iuneq2i.1 | ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) | 
| Ref | Expression | 
|---|---|
| iineq2i | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | iineq2 5011 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) | |
| 2 | iuneq2i.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) | |
| 3 | 1, 2 | mprg 3066 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∩ ciin 4991 | 
| 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-8 2109 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-clel 2815 df-ral 3061 df-iin 4993 | 
| This theorem is referenced by: iinrab 5068 iinin1 5078 diaintclN 41061 dibintclN 41170 dihintcl 41347 imaiinfv 42709 smflimlem3 46793 | 
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