Theorem csbeq1 3150

Description: Analog of dfsbcq 3053 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbeq1	⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)

Proof of Theorem csbeq1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 3053	. . 3 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐵 / 𝑥]𝑦 ∈ 𝐶))
2	1	abbidv 2358	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶})
3		df-csb 3148	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
4		df-csb 3148	. 2 ⊢ ⦋𝐵 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶}
5	2, 3, 4	3eqtr4g 2296	1 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  {cab 2224  [wsbc 3051  ⦋csb 3147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052  df-csb 3148

This theorem is used by:  csbeq1d  3154  csbeq1a  3156  csbiebg  3190  sbcnestgf  3199  cbvralcsf  3210  cbvrexcsf  3211  cbvreucsf  3212  cbvrabcsf  3213  csbing  3438  ifeqeqxdc  3687  disjnims  4121  sbcbrg  4186  csbopabg  4210  pofun  4459  csbima12g  5151  csbiotag  5373  fvmpts  5786  fvmpt2  5792  mptfvex  5794  elfvmptrab1  5804  fmptcof  5878  fmptcos  5879  fliftfuns  6007  csbriotag  6055  riotaeqimp  6066  csbov123g  6127  elovmporab1w  6293  eqerlem  6842  qliftfuns  6897  summodclem2a  12174  zsumdc  12177  fsum3  12180  sumsnf  12202  sumsns  12208  fsum2dlemstep  12227  fisumcom2  12231  fsumshftm  12238  fisum0diag2  12240  fsumiun  12270  prodsnf  12385  fprodm1s  12394  fprodp1s  12395  prodsns  12396  fprod2dlemstep  12415  fprodcom2fi  12419  pcmptdvds  13154  ctiunctlemf  13388  psrass1filem  15182  mulcncflem  15845  fsumdvdsmul  16297