Theorem csbeq1d 3087

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 3083	. 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 3080

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 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-sbc 2986  df-csb 3081

This theorem is referenced by:  csbidmg  3137  csbco3g  3139  fmptcof  5725  mpomptsx  6250  dmmpossx  6252  fmpox  6253  fmpoco  6269  xpf1o  6900  summodclem3  11523  summodclem2a  11524  summodc  11526  zsumdc  11527  fsum3  11530  sumsnf  11552  fsumcnv  11580  fisumcom2  11581  fsumshftm  11588  fisum0diag2  11590  prodmodclem3  11718  prodmodclem2a  11719  prodmodc  11721  zproddc  11722  fprodseq  11726  prodsnf  11735  fprodcnv  11768  fprodcom2fi  11769  pcmpt  12481  ctiunctlemu1st  12591  ctiunctlemu2nd  12592  ctiunctlemudc  12594  ctiunctlemfo  12596  prdsex  12880  imasex  12888  psrval  14152