Theorem csbeq1d 3148

Description: Equality deduction for proper substitution into a class. (Contributed by NM, 3-Dec-2005.)

Hypothesis

Ref	Expression
csbeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
csbeq1d	⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)

Proof of Theorem csbeq1d

Step	Hyp	Ref	Expression
1		csbeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		csbeq1 3144	. 2 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)
3	1, 2	syl 14	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)

Syntax hints:   → wi 4   = wceq 1398  ⦋csb 3141

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-sbc 3046  df-csb 3142

This theorem is referenced by:  csbidmg  3198  csbco3g  3200  fmptcof  5849  mpomptsx  6406  dmmpossx  6408  fmpox  6409  fmpoco  6425  xpf1o  7110  summodclem3  12094  summodclem2a  12095  summodc  12097  zsumdc  12098  fsum3  12101  sumsnf  12123  fsumcnv  12151  fisumcom2  12152  fsumshftm  12159  fisum0diag2  12161  prodmodclem3  12289  prodmodclem2a  12290  prodmodc  12292  zproddc  12293  fprodseq  12297  prodsnf  12306  fprodcnv  12339  fprodcom2fi  12340  pcmpt  13069  ctiunctlemu1st  13272  ctiunctlemu2nd  13273  ctiunctlemudc  13275  ctiunctlemfo  13277  imasex  13572  prdsex  14117  psrval  14943  fsumdvdsmul  15988