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Mirrors > Home > ILE Home > Th. List > 3eltr3d | GIF version |
Description: Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
3eltr3d.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
3eltr3d.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
3eltr3d.3 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
3eltr3d | ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eltr3d.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
2 | 3eltr3d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
3 | 3eltr3d.3 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
4 | 2, 3 | eleqtrd 2245 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
5 | 1, 4 | eqeltrrd 2244 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1343 ∈ wcel 2136 |
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 1435 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-4 1498 ax-17 1514 ax-ial 1522 ax-ext 2147 |
This theorem depends on definitions: df-bi 116 df-cleq 2158 df-clel 2161 |
This theorem is referenced by: reg3exmidlemwe 4556 nnaordi 6476 icoshftf1o 9927 lincmb01cmp 9939 fzosubel 10129 cnmpt2res 12947 dvcnp2cntop 13313 |
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