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Mirrors > Home > ILE Home > Th. List > iineq2i | 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 3878 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶) | |
2 | iuneq2i.1 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) | |
3 | 1, 2 | mprg 2521 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ 𝐴 𝐶 |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ∩ ciin 3862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-ral 2447 df-iin 3864 |
This theorem is referenced by: iinrabm 3923 iinin1m 3930 |
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