Theorem csbeq1a 2916

Description: Equality theorem for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbeq1a	⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵)

Proof of Theorem csbeq1a

Step	Hyp	Ref	Expression
1		csbid 2915	. 2 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵
2		csbeq1 2911	. 2 ⊢ (𝑥 = 𝐴 → ⦋𝑥 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3	1, 2	syl5eqr 2127	1 ⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   = wceq 1284  ⦋csb 2908

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 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-sbc 2816  df-csb 2909

This theorem is referenced by:  csbhypf  2941  csbiebt  2942  sbcnestgf  2953  cbvralcsf  2964  cbvrexcsf  2965  cbvreucsf  2966  cbvrabcsf  2967  csbing  3173  sbcbrg  3834  moop2  4006  pofun  4067  eusvnf  4203  opeliunxp  4413  elrnmpt1  4603  csbima12g  4706  fvmpts  5271  fvmpt2  5275  mptfvex  5277  fmptco  5351  fmptcof  5352  fmptcos  5353  elabrex  5418  fliftfuns  5458  csbov123g  5563  ovmpt2s  5644  csbopeq1a  5834  mpt2mptsx  5843  dmmpt2ssx  5845  fmpt2x  5846  mpt2fvex  5849  fmpt2co  5857  eqerlem  6160  qliftfuns  6213