Theorem csbeq1 3127

Description: Analog of dfsbcq 3030 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 3030	. . 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 3125	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
4		df-csb 3125	. 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 3028  ⦋csb 3124

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 3029  df-csb 3125

This theorem is referenced by:  csbeq1d  3131  csbeq1a  3133  csbiebg  3167  sbcnestgf  3176  cbvralcsf  3187  cbvrexcsf  3188  cbvreucsf  3189  cbvrabcsf  3190  csbing  3411  disjnims  4074  sbcbrg  4138  csbopabg  4162  pofun  4403  csbima12g  5089  csbiotag  5311  fvmpts  5714  fvmpt2  5720  mptfvex  5722  elfvmptrab1  5731  fmptcof  5804  fmptcos  5805  fliftfuns  5928  csbriotag  5974  riotaeqimp  5985  csbov123g  6046  elovmporab1w  6212  eqerlem  6719  qliftfuns  6774  summodclem2a  11900  zsumdc  11903  fsum3  11906  sumsnf  11928  sumsns  11934  fsum2dlemstep  11953  fisumcom2  11957  fsumshftm  11964  fisum0diag2  11966  fsumiun  11996  prodsnf  12111  fprodm1s  12120  fprodp1s  12121  prodsns  12122  fprod2dlemstep  12141  fprodcom2fi  12145  pcmptdvds  12876  ctiunctlemf  13017  mulcncflem  15289  fsumdvdsmul  15673