Theorem csbeq1 3087

Description: Analog of dfsbcq 2991 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 2991	. . 3 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐵 / 𝑥]𝑦 ∈ 𝐶))
2	1	abbidv 2314	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶})
3		df-csb 3085	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
4		df-csb 3085	. 2 ⊢ ⦋𝐵 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶}
5	2, 3, 4	3eqtr4g 2254	1 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  {cab 2182  [wsbc 2989  ⦋csb 3084

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-sbc 2990  df-csb 3085

This theorem is referenced by:  csbeq1d  3091  csbeq1a  3093  csbiebg  3127  sbcnestgf  3136  cbvralcsf  3147  cbvrexcsf  3148  cbvreucsf  3149  cbvrabcsf  3150  csbing  3370  disjnims  4025  sbcbrg  4087  csbopabg  4111  pofun  4347  csbima12g  5030  csbiotag  5251  fvmpts  5639  fvmpt2  5645  mptfvex  5647  elfvmptrab1  5656  fmptcof  5729  fmptcos  5730  fliftfuns  5845  csbriotag  5890  csbov123g  5960  elovmporab1w  6124  eqerlem  6623  qliftfuns  6678  summodclem2a  11546  zsumdc  11549  fsum3  11552  sumsnf  11574  sumsns  11580  fsum2dlemstep  11599  fisumcom2  11603  fsumshftm  11610  fisum0diag2  11612  fsumiun  11642  prodsnf  11757  fprodm1s  11766  fprodp1s  11767  prodsns  11768  fprod2dlemstep  11787  fprodcom2fi  11791  pcmptdvds  12514  ctiunctlemf  12655  mulcncflem  14843  fsumdvdsmul  15227