Theorem csbeq1 3128

Description: Analog of dfsbcq 3031 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 3031	. . 3 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐵 / 𝑥]𝑦 ∈ 𝐶))
2	1	abbidv 2347	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶})
3		df-csb 3126	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
4		df-csb 3126	. 2 ⊢ ⦋𝐵 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶}
5	2, 3, 4	3eqtr4g 2287	1 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  {cab 2215  [wsbc 3029  ⦋csb 3125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3030  df-csb 3126

This theorem is referenced by:  csbeq1d  3132  csbeq1a  3134  csbiebg  3168  sbcnestgf  3177  cbvralcsf  3188  cbvrexcsf  3189  cbvreucsf  3190  cbvrabcsf  3191  csbing  3412  disjnims  4077  sbcbrg  4141  csbopabg  4165  pofun  4407  csbima12g  5095  csbiotag  5317  fvmpts  5720  fvmpt2  5726  mptfvex  5728  elfvmptrab1  5737  fmptcof  5810  fmptcos  5811  fliftfuns  5934  csbriotag  5980  riotaeqimp  5991  csbov123g  6052  elovmporab1w  6218  eqerlem  6728  qliftfuns  6783  summodclem2a  11935  zsumdc  11938  fsum3  11941  sumsnf  11963  sumsns  11969  fsum2dlemstep  11988  fisumcom2  11992  fsumshftm  11999  fisum0diag2  12001  fsumiun  12031  prodsnf  12146  fprodm1s  12155  fprodp1s  12156  prodsns  12157  fprod2dlemstep  12176  fprodcom2fi  12180  pcmptdvds  12911  ctiunctlemf  13052  mulcncflem  15324  fsumdvdsmul  15708