Theorem csbeq1d 3144

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 3140	. 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 3137

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-sbc 3042  df-csb 3138

This theorem is referenced by:  csbidmg  3194  csbco3g  3196  fmptcof  5843  mpomptsx  6392  dmmpossx  6394  fmpox  6395  fmpoco  6411  xpf1o  7096  summodclem3  12059  summodclem2a  12060  summodc  12062  zsumdc  12063  fsum3  12066  sumsnf  12088  fsumcnv  12116  fisumcom2  12117  fsumshftm  12124  fisum0diag2  12126  prodmodclem3  12254  prodmodclem2a  12255  prodmodc  12257  zproddc  12258  fprodseq  12262  prodsnf  12271  fprodcnv  12304  fprodcom2fi  12305  pcmpt  13034  ctiunctlemu1st  13174  ctiunctlemu2nd  13175  ctiunctlemudc  13177  ctiunctlemfo  13179  prdsex  13471  imasex  13507  psrval  14801  fsumdvdsmul  15846