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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 2161 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶) |
4 | 1, 3 | mpbi 144 | 1 ⊢ 𝐴 ∈ 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1296 ∈ wcel 1445 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1388 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-4 1452 ax-17 1471 ax-ial 1479 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-cleq 2088 df-clel 2091 |
This theorem is referenced by: eleqtrri 2170 3eltr3i 2175 prid2 3569 2eluzge0 9162 fz01or 9674 ef0lem 11099 ege2le3 11110 efgt1p2 11134 efgt1p 11135 phi1 11622 |
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