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Mirrors > Home > ILE Home > Th. List > 3eltr4i | GIF version |
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
3eltr4.1 | ⊢ 𝐴 ∈ 𝐵 |
3eltr4.2 | ⊢ 𝐶 = 𝐴 |
3eltr4.3 | ⊢ 𝐷 = 𝐵 |
Ref | Expression |
---|---|
3eltr4i | ⊢ 𝐶 ∈ 𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eltr4.2 | . 2 ⊢ 𝐶 = 𝐴 | |
2 | 3eltr4.1 | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
3 | 3eltr4.3 | . . 3 ⊢ 𝐷 = 𝐵 | |
4 | 2, 3 | eleqtrri 2193 | . 2 ⊢ 𝐴 ∈ 𝐷 |
5 | 1, 4 | eqeltri 2190 | 1 ⊢ 𝐶 ∈ 𝐷 |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∈ wcel 1465 |
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 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-17 1491 ax-ial 1499 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-cleq 2110 df-clel 2113 |
This theorem is referenced by: 1nq 7142 0r 7526 1sr 7527 m1r 7528 |
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