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Mirrors > Home > ILE Home > Th. List > eleqtri | GIF version |
Description: Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
eleqtr.1 | ⊢ 𝐴 ∈ 𝐵 |
eleqtr.2 | ⊢ 𝐵 = 𝐶 |
Ref | Expression |
---|---|
eleqtri | ⊢ 𝐴 ∈ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleqtr.1 | . 2 ⊢ 𝐴 ∈ 𝐵 | |
2 | eleqtr.2 | . . 3 ⊢ 𝐵 = 𝐶 | |
3 | 2 | eleq2i 2256 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶) |
4 | 1, 3 | mpbi 145 | 1 ⊢ 𝐴 ∈ 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-cleq 2182 df-clel 2185 |
This theorem is referenced by: eleqtrri 2265 3eltr3i 2270 prid2 3714 2eluzge0 9605 fz01or 10141 fz0to4untppr 10154 ef0lem 11700 ege2le3 11711 efgt1p2 11735 efgt1p 11736 phi1 12251 cnrehmeocntop 14553 dvcjbr 14632 fmelpw1o 15019 |
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