Theorem csbeq1 3084

Description: Analog of dfsbcq 2988 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 2988	. . 3 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐵 / 𝑥]𝑦 ∈ 𝐶))
2	1	abbidv 2311	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶})
3		df-csb 3082	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
4		df-csb 3082	. 2 ⊢ ⦋𝐵 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶}
5	2, 3, 4	3eqtr4g 2251	1 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  {cab 2179  [wsbc 2986  ⦋csb 3081

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-sbc 2987  df-csb 3082

This theorem is referenced by:  csbeq1d  3088  csbeq1a  3090  csbiebg  3124  sbcnestgf  3133  cbvralcsf  3144  cbvrexcsf  3145  cbvreucsf  3146  cbvrabcsf  3147  csbing  3367  disjnims  4022  sbcbrg  4084  csbopabg  4108  pofun  4344  csbima12g  5027  csbiotag  5248  fvmpts  5636  fvmpt2  5642  mptfvex  5644  elfvmptrab1  5653  fmptcof  5726  fmptcos  5727  fliftfuns  5842  csbriotag  5887  csbov123g  5957  elovmporab1w  6121  eqerlem  6620  qliftfuns  6675  summodclem2a  11527  zsumdc  11530  fsum3  11533  sumsnf  11555  sumsns  11561  fsum2dlemstep  11580  fisumcom2  11584  fsumshftm  11591  fisum0diag2  11593  fsumiun  11623  prodsnf  11738  fprodm1s  11747  fprodp1s  11748  prodsns  11749  fprod2dlemstep  11768  fprodcom2fi  11772  pcmptdvds  12486  ctiunctlemf  12598  mulcncflem  14786