Theorem csbeq1d 3052

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 3048	. 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 1343  ⦋csb 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-sbc 2952  df-csb 3046

This theorem is referenced by:  csbidmg  3101  csbco3g  3103  fmptcof  5652  mpomptsx  6165  dmmpossx  6167  fmpox  6168  fmpoco  6184  xpf1o  6810  summodclem3  11321  summodclem2a  11322  summodc  11324  zsumdc  11325  fsum3  11328  sumsnf  11350  fsumcnv  11378  fisumcom2  11379  fsumshftm  11386  fisum0diag2  11388  prodmodclem3  11516  prodmodclem2a  11517  prodmodc  11519  zproddc  11520  fprodseq  11524  prodsnf  11533  fprodcnv  11566  fprodcom2fi  11567  pcmpt  12273  ctiunctlemu1st  12367  ctiunctlemu2nd  12368  ctiunctlemudc  12370  ctiunctlemfo  12372