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Mirrors > Home > ILE Home > Th. List > eleqtrrid | GIF version |
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
Ref | Expression |
---|---|
eleqtrrid.1 | ⊢ 𝐴 ∈ 𝐵 |
eleqtrrid.2 | ⊢ (𝜑 → 𝐶 = 𝐵) |
Ref | Expression |
---|---|
eleqtrrid | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleqtrrid.1 | . 2 ⊢ 𝐴 ∈ 𝐵 | |
2 | eleqtrrid.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐵) | |
3 | 2 | eqcomd 2176 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) |
4 | 1, 3 | eleqtrid 2259 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 |
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 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-17 1519 ax-ial 1527 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-cleq 2163 df-clel 2166 |
This theorem is referenced by: rabsnt 3656 0elnn 4601 canth 5805 tfrexlem 6311 rdgtfr 6351 rdgruledefgg 6352 exmidonfinlem 7163 hashinfom 10705 ennnfonelemhom 12363 exmid1stab 13998 |
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