Theorem csbeq1a 2917

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 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)

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