Theorem csbeq1 3150

Description: Analog of dfsbcq 3053 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 3053	. . 3 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐵 / 𝑥]𝑦 ∈ 𝐶))
2	1	abbidv 2358	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶})
3		df-csb 3148	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶}
4		df-csb 3148	. 2 ⊢ ⦋𝐵 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶}
5	2, 3, 4	3eqtr4g 2296	1 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  {cab 2224  [wsbc 3051  ⦋csb 3147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052  df-csb 3148

This theorem is used by:  csbeq1d  3154  csbeq1a  3156  csbiebg  3190  sbcnestgf  3199  cbvralcsf  3210  cbvrexcsf  3211  cbvreucsf  3212  cbvrabcsf  3213  csbing  3438  ifeqeqxdc  3687  disjnims  4121  sbcbrg  4185  csbopabg  4209  pofun  4457  csbima12g  5148  csbiotag  5370  fvmpts  5783  fvmpt2  5789  mptfvex  5791  elfvmptrab1  5801  fmptcof  5875  fmptcos  5876  fliftfuns  6004  csbriotag  6052  riotaeqimp  6063  csbov123g  6124  elovmporab1w  6290  eqerlem  6838  qliftfuns  6893  summodclem2a  12150  zsumdc  12153  fsum3  12156  sumsnf  12178  sumsns  12184  fsum2dlemstep  12203  fisumcom2  12207  fsumshftm  12214  fisum0diag2  12216  fsumiun  12246  prodsnf  12361  fprodm1s  12370  fprodp1s  12371  prodsns  12372  fprod2dlemstep  12391  fprodcom2fi  12395  pcmptdvds  13126  ctiunctlemf  13331  mulcncflem  15710  fsumdvdsmul  16111