Theorem csbeq1d 3154

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 3150	. 2 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)
3	1, 2	syl 14	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶)

This proof depends on syntax axioms:   → wi 4   = wceq 1402  ⦋csb 3147

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-sbc 3052  df-csb 3148

This theorem is used by:  csbidmg  3204  csbco3g  3206  fmptcof  5875  mpomptsx  6433  dmmpossx  6435  fmpox  6436  fmpoco  6452  xpf1o  7144  summodclem3  12149  summodclem2a  12150  summodc  12152  zsumdc  12153  fsum3  12156  sumsnf  12178  fsumcnv  12206  fisumcom2  12207  fsumshftm  12214  fisum0diag2  12216  prodmodclem3  12344  prodmodclem2a  12345  prodmodc  12347  zproddc  12348  fprodseq  12352  prodsnf  12361  fprodcnv  12394  fprodcom2fi  12395  pcmpt  13124  ctiunctlemu1st  13327  ctiunctlemu2nd  13328  ctiunctlemudc  13330  ctiunctlemfo  13332  imasex  13628  prdsex  14174  psrval  15052  fsumdvdsmul  16111