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Mirrors > Home > ILE Home > Th. List > 3eltr4d | GIF version |
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
3eltr4d.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
3eltr4d.2 | ⊢ (𝜑 → 𝐶 = 𝐴) |
3eltr4d.3 | ⊢ (𝜑 → 𝐷 = 𝐵) |
Ref | Expression |
---|---|
3eltr4d | ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eltr4d.2 | . 2 ⊢ (𝜑 → 𝐶 = 𝐴) | |
2 | 3eltr4d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
3 | 3eltr4d.3 | . . 3 ⊢ (𝜑 → 𝐷 = 𝐵) | |
4 | 2, 3 | eleqtrrd 2158 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
5 | 1, 4 | eqeltrd 2155 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 df-clel 2077 |
This theorem is referenced by: ovmpt2dxf 5646 nnaordi 6104 iccf1o 9026 |
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