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Mirrors > Home > MPE Home > Th. List > iineq2i | Structured version Visualization version GIF version |
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 5017 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) | |
2 | iuneq2i.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) | |
3 | 1, 2 | mprg 3056 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∩ ciin 4998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-iin 5000 |
This theorem is referenced by: iinrab 5073 iinin1 5083 diaintclN 40661 dibintclN 40770 dihintcl 40947 imaiinfv 42255 smflimlem3 46299 |
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