Theorem csbeq1d 3079

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 3075	. 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 1364  ⦋csb 3072

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-sbc 2978  df-csb 3073

This theorem is referenced by:  csbidmg  3128  csbco3g  3130  fmptcof  5699  mpomptsx  6216  dmmpossx  6218  fmpox  6219  fmpoco  6235  xpf1o  6862  summodclem3  11406  summodclem2a  11407  summodc  11409  zsumdc  11410  fsum3  11413  sumsnf  11435  fsumcnv  11463  fisumcom2  11464  fsumshftm  11471  fisum0diag2  11473  prodmodclem3  11601  prodmodclem2a  11602  prodmodc  11604  zproddc  11605  fprodseq  11609  prodsnf  11618  fprodcnv  11651  fprodcom2fi  11652  pcmpt  12359  ctiunctlemu1st  12453  ctiunctlemu2nd  12454  ctiunctlemudc  12456  ctiunctlemfo  12458  prdsex  12740  imasex  12748