Theorem csbeq1a 2860

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 2859	. 2 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵
2		csbeq1 2855	. 2 ⊢ (𝑥 = 𝐴 → ⦋𝑥 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
3	1, 2	syl5eqr 2086	1 ⊢ (𝑥 = 𝐴 → 𝐵 = ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   = wceq 1243  ⦋csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765  df-csb 2853

This theorem is referenced by:  csbhypf  2885  csbiebt  2886  sbcnestgf  2897  cbvralcsf  2908  cbvrexcsf  2909  cbvreucsf  2910  cbvrabcsf  2911  csbing  3144  sbcbrg  3813  moop2  3988  pofun  4049  eusvnf  4185  opeliunxp  4395  elrnmpt1  4585  csbima12g  4686  fvmpts  5250  fvmpt2  5254  mptfvex  5256  fmptco  5330  fmptcof  5331  fmptcos  5332  elabrex  5397  fliftfuns  5438  csbov123g  5543  ovmpt2s  5624  csbopeq1a  5814  mpt2mptsx  5823  dmmpt2ssx  5825  fmpt2x  5826  mpt2fvex  5829  fmpt2co  5837  eqerlem  6137  qliftfuns  6190