Theorem csbeq1 3075

Description: Analog of dfsbcq 2979 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 2979	. . 3 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐵 / 𝑥]𝑦 ∈ 𝐶))
2	1	abbidv 2307	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶})
3		df-csb 3073	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
4		df-csb 3073	. 2 ⊢ ⦋𝐵 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶}
5	2, 3, 4	3eqtr4g 2247	1 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2160  {cab 2175  [wsbc 2977  ⦋csb 3072

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-sbc 2978  df-csb 3073

This theorem is referenced by:  csbeq1d  3079  csbeq1a  3081  csbiebg  3114  sbcnestgf  3123  cbvralcsf  3134  cbvrexcsf  3135  cbvreucsf  3136  cbvrabcsf  3137  csbing  3357  disjnims  4010  sbcbrg  4072  csbopabg  4096  pofun  4327  csbima12g  5004  csbiotag  5225  fvmpts  5611  fvmpt2  5616  mptfvex  5618  elfvmptrab1  5627  fmptcof  5700  fmptcos  5701  fliftfuns  5816  csbriotag  5860  csbov123g  5930  eqerlem  6585  qliftfuns  6638  summodclem2a  11409  zsumdc  11412  fsum3  11415  sumsnf  11437  sumsns  11443  fsum2dlemstep  11462  fisumcom2  11466  fsumshftm  11473  fisum0diag2  11475  fsumiun  11505  prodsnf  11620  fprodm1s  11629  fprodp1s  11630  prodsns  11631  fprod2dlemstep  11650  fprodcom2fi  11654  pcmptdvds  12363  ctiunctlemf  12464  mulcncflem  14502