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  3371  disjnims  4026  sbcbrg  4088  csbopabg  4112  pofun  4348  csbima12g  5031  csbiotag  5252  fvmpts  5642  fvmpt2  5648  mptfvex  5650  elfvmptrab1  5659  fmptcof  5732  fmptcos  5733  fliftfuns  5848  csbriotag  5893  csbov123g  5964  elovmporab1w  6128  eqerlem  6632  qliftfuns  6687  summodclem2a  11563  zsumdc  11566  fsum3  11569  sumsnf  11591  sumsns  11597  fsum2dlemstep  11616  fisumcom2  11620  fsumshftm  11627  fisum0diag2  11629  fsumiun  11659  prodsnf  11774  fprodm1s  11783  fprodp1s  11784  prodsns  11785  fprod2dlemstep  11804  fprodcom2fi  11808  pcmptdvds  12539  ctiunctlemf  12680  mulcncflem  14927  fsumdvdsmul  15311