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Mirrors > Home > ILE Home > Th. List > 3eltr3i | GIF version |
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
3eltr3.1 | ⊢ 𝐴 ∈ 𝐵 |
3eltr3.2 | ⊢ 𝐴 = 𝐶 |
3eltr3.3 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
3eltr3i | ⊢ 𝐶 ∈ 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eltr3.2 | . 2 ⊢ 𝐴 = 𝐶 | |
2 | 3eltr3.1 | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
3 | 3eltr3.3 | . . 3 ⊢ 𝐵 = 𝐷 | |
4 | 2, 3 | eleqtri 2264 | . 2 ⊢ 𝐴 ∈ 𝐷 |
5 | 1, 4 | eqeltrri 2263 | 1 ⊢ 𝐶 ∈ 𝐷 |
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: (None) |
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