Theorem csbeq1d 3076

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 3072	. 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 1363  ⦋csb 3069

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-sbc 2975  df-csb 3070

This theorem is referenced by:  csbidmg  3125  csbco3g  3127  fmptcof  5696  mpomptsx  6211  dmmpossx  6213  fmpox  6214  fmpoco  6230  xpf1o  6857  summodclem3  11401  summodclem2a  11402  summodc  11404  zsumdc  11405  fsum3  11408  sumsnf  11430  fsumcnv  11458  fisumcom2  11459  fsumshftm  11466  fisum0diag2  11468  prodmodclem3  11596  prodmodclem2a  11597  prodmodc  11599  zproddc  11600  fprodseq  11604  prodsnf  11613  fprodcnv  11646  fprodcom2fi  11647  pcmpt  12354  ctiunctlemu1st  12448  ctiunctlemu2nd  12449  ctiunctlemudc  12451  ctiunctlemfo  12453  prdsex  12735  imasex  12743