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Mirrors > Home > ILE Home > Th. List > eleqtrid | GIF version |
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
Ref | Expression |
---|---|
eleqtrid.1 | ⊢ 𝐴 ∈ 𝐵 |
eleqtrid.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
eleqtrid | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleqtrid.1 | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
2 | 1 | a1i 9 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
3 | eleqtrid.2 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) | |
4 | 2, 3 | eleqtrd 2219 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 |
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 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-17 1507 ax-ial 1515 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-cleq 2133 df-clel 2136 |
This theorem is referenced by: eleqtrrid 2230 opth1 4166 opth 4167 eqelsuc 4349 txdis 12485 bj-nnelirr 13322 |
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