Theorem eleq1i 2295

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 2292	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)

Syntax hints:   ↔ wb 105   = wceq 1395   ∈ wcel 2200

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  eleq12i  2297  eqeltri  2302  intexrabim  4236  abssexg  4265  abnex  4537  snnex  4538  pwexb  4564  sucexb  4588  omex  4684  iprc  4992  dfse2  5100  fressnfv  5825  fnotovb  6046  f1stres  6303  f2ndres  6304  ottposg  6399  dftpos4  6407  frecabex  6542  oacl  6604  diffifi  7052  djuexb  7207  pitonn  8031  axicn  8046  pnfnre  8184  mnfnre  8185  0mnnnnn0  9397  pfxccatin12lem3  11259  pfxccat3  11261  swrdccat  11262  pfxccat3a  11265  swrdccat3blem  11266  swrdccat3b  11267  nprmi  12641  issubm  13500  issrg  13923  srgfcl  13931  subrngrng  14160  txdis1cn  14946  xmeterval  15103  expcncf  15277  gausslemma2dlem1a  15731  2lgslem4  15776  bj-sucexg  16243