Theorem eqeq2i 2099

Description: Inference from equality to equivalence of equalities. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
eqeq2i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
eqeq2i	⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)

Proof of Theorem eqeq2i

Step	Hyp	Ref	Expression
1		eqeq2i.1	. 2 ⊢ 𝐴 = 𝐵
2		eqeq2 2098	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵))
3	1, 2	ax-mp 7	1 ⊢ (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)

Syntax hints:   ↔ wb 104   = wceq 1290

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-4 1446  ax-17 1465  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-cleq 2082

This theorem is referenced by:  eqtri  2109  rabid2  2544  dfss2  3015  equncom  3146  preq12b  3620  preqsn  3625  opeqpr  4089  orddif  4376  dfrel4v  4895  dfiota2  4994  funopg  5061  fnressn  5497  fressnfv  5498  acexmidlemph  5659  fnovim  5767  tpossym  6055  qsid  6371  mapsncnv  6466  ixpsnf1o  6507  unfiexmid  6682  recidpirq  7456  axprecex  7476  negeq0  7797  muleqadd  8198  fihasheq0  10263  cjne0  10403  sqrt00  10534  sqrtmsq2i  10629  cbvsum  10810  fsump1i  10888  mertenslem2  10991  absefib  11121  efieq1re  11122