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Mirrors > Home > ILE Home > Th. List > eqeltrrid | GIF version |
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
Ref | Expression |
---|---|
eqeltrrid.1 | ⊢ 𝐵 = 𝐴 |
eqeltrrid.2 | ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
Ref | Expression |
---|---|
eqeltrrid | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeltrrid.1 | . . 3 ⊢ 𝐵 = 𝐴 | |
2 | 1 | eqcomi 2161 | . 2 ⊢ 𝐴 = 𝐵 |
3 | eqeltrrid.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶) | |
4 | 2, 3 | eqeltrid 2244 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∈ wcel 2128 |
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 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-17 1506 ax-ial 1514 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-cleq 2150 df-clel 2153 |
This theorem is referenced by: dmrnssfld 4851 cnvexg 5125 opabbrex 5867 offval 6041 resfunexgALT 6060 abrexexg 6068 abrexex2g 6070 opabex3d 6071 oprssdmm 6121 unfidisj 6868 ssfii 6920 djuexb 6990 nqprlu 7469 iccshftr 9904 iccshftl 9906 iccdil 9908 icccntr 9910 mertenslem2 11444 exprmfct 12030 ennnfonelemg 12202 0opn 12474 difopn 12578 tgrest 12639 txbasex 12727 txdis1cn 12748 cnmptid 12751 cnmptc 12752 cnmpt1st 12758 cnmpt2nd 12759 cnmpt2c 12760 hmeoima 12780 hmeocld 12782 fsumcncntop 13026 |
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