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Mirrors > Home > ILE Home > Th. List > csbeq1a | GIF version |
Description: Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
Ref | Expression |
---|---|
csbeq1a | ⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbid 2916 | . 2 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵 | |
2 | csbeq1 2912 | . 2 ⊢ (𝑥 = 𝐴 → ⦋𝑥 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵) | |
3 | 1, 2 | syl5eqr 2128 | 1 ⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ⦋csb 2909 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-11 1438 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-sbc 2817 df-csb 2910 |
This theorem is referenced by: csbhypf 2942 csbiebt 2943 sbcnestgf 2954 cbvralcsf 2965 cbvrexcsf 2966 cbvreucsf 2967 cbvrabcsf 2968 csbing 3174 sbcbrg 3836 moop2 4008 pofun 4069 eusvnf 4205 opeliunxp 4415 elrnmpt1 4607 csbima12g 4710 fvmpts 5276 fvmpt2 5280 mptfvex 5282 fmptco 5356 fmptcof 5357 fmptcos 5358 elabrex 5423 fliftfuns 5463 csbov123g 5568 ovmpt2s 5649 csbopeq1a 5839 mpt2mptsx 5848 dmmpt2ssx 5850 fmpt2x 5851 mpt2fvex 5854 fmpt2co 5862 eqerlem 6196 qliftfuns 6249 xpf1o 6375 sumeq2d 10323 sumeq2 10324 |
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