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Mirrors > Home > ILE Home > Th. List > eqeltrrdi | GIF version |
Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
Ref | Expression |
---|---|
eqeltrrdi.1 | ⊢ (𝜑 → 𝐵 = 𝐴) |
eqeltrrdi.2 | ⊢ 𝐵 ∈ 𝐶 |
Ref | Expression |
---|---|
eqeltrrdi | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeltrrdi.1 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐴) | |
2 | 1 | eqcomd 2145 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) |
3 | eqeltrrdi.2 | . 2 ⊢ 𝐵 ∈ 𝐶 | |
4 | 2, 3 | eqeltrdi 2230 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 |
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 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-cleq 2132 df-clel 2135 |
This theorem is referenced by: eusvnfb 4375 releldm2 6083 mapprc 6546 ixpprc 6613 ixpssmap2g 6621 ixpssmapg 6622 bren 6641 brdomg 6642 mapen 6740 ssenen 6745 fi0 6863 ioof 9757 hashfacen 10582 fsum3 11159 cnrehmeocntop 12765 |
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