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  6843  qliftfuns  6898  summodclem2a  12172  zsumdc  12175  fsum3  12178  sumsnf  12200  sumsns  12206  fsum2dlemstep  12225  fisumcom2  12229  fsumshftm  12236  fisum0diag2  12238  fsumiun  12268  prodsnf  12383  fprodm1s  12392  fprodp1s  12393  prodsns  12394  fprod2dlemstep  12413  fprodcom2fi  12417  pcmptdvds  13152  ctiunctlemf  13386  psrass1filem  15186  mulcncflem  15850  fsumdvdsmul  16302