Theorem csbeq1d 3047

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 3043	. 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 1342  ⦋csb 3040

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-sbc 2947  df-csb 3041

This theorem is referenced by:  csbidmg  3096  csbco3g  3098  fmptcof  5646  mpomptsx  6157  dmmpossx  6159  fmpox  6160  fmpoco  6175  xpf1o  6801  summodclem3  11307  summodclem2a  11308  summodc  11310  zsumdc  11311  fsum3  11314  sumsnf  11336  fsumcnv  11364  fisumcom2  11365  fsumshftm  11372  fisum0diag2  11374  prodmodclem3  11502  prodmodclem2a  11503  prodmodc  11505  zproddc  11506  fprodseq  11510  prodsnf  11519  fprodcnv  11552  fprodcom2fi  11553  pcmpt  12250  ctiunctlemu1st  12304  ctiunctlemu2nd  12305  ctiunctlemudc  12307  ctiunctlemfo  12309