Theorem eleq1i 2297

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

Hypothesis

Ref	Expression
eleq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
eleq1i	⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)

Proof of Theorem eleq1i

Step	Hyp	Ref	Expression
1		eleq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		eleq1 2294	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)

Syntax hints:   ↔ wb 105   = wceq 1397   ∈ wcel 2202

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227

This theorem is referenced by:  eleq12i  2299  eqeltri  2304  intexrabim  4243  abssexg  4272  abnex  4544  snnex  4545  pwexb  4571  sucexb  4595  omex  4691  iprc  5001  dfse2  5109  fressnfv  5840  fnotovb  6063  f1stres  6321  f2ndres  6322  ottposg  6420  dftpos4  6428  frecabex  6563  oacl  6627  diffifi  7082  djuexb  7242  pitonn  8067  axicn  8082  pnfnre  8220  mnfnre  8221  0mnnnnn0  9433  pfxccatin12lem3  11312  pfxccat3  11314  swrdccat  11315  pfxccat3a  11318  swrdccat3blem  11319  swrdccat3b  11320  nprmi  12695  issubm  13554  issrg  13977  srgfcl  13985  subrngrng  14215  txdis1cn  15001  xmeterval  15158  expcncf  15332  gausslemma2dlem1a  15786  2lgslem4  15831  clwwlknonex2  16289  bj-sucexg  16517