Theorem csbeq1d 3131

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 3127	. 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 1395  ⦋csb 3124

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-sbc 3029  df-csb 3125

This theorem is referenced by:  csbidmg  3181  csbco3g  3183  fmptcof  5804  mpomptsx  6349  dmmpossx  6351  fmpox  6352  fmpoco  6368  xpf1o  7013  summodclem3  11899  summodclem2a  11900  summodc  11902  zsumdc  11903  fsum3  11906  sumsnf  11928  fsumcnv  11956  fisumcom2  11957  fsumshftm  11964  fisum0diag2  11966  prodmodclem3  12094  prodmodclem2a  12095  prodmodc  12097  zproddc  12098  fprodseq  12102  prodsnf  12111  fprodcnv  12144  fprodcom2fi  12145  pcmpt  12874  ctiunctlemu1st  13013  ctiunctlemu2nd  13014  ctiunctlemudc  13016  ctiunctlemfo  13018  prdsex  13310  imasex  13346  psrval  14638  fsumdvdsmul  15673