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  4237  abssexg  4266  abnex  4538  snnex  4539  pwexb  4565  sucexb  4589  omex  4685  iprc  4993  dfse2  5101  fressnfv  5830  fnotovb  6053  f1stres  6311  f2ndres  6312  ottposg  6407  dftpos4  6415  frecabex  6550  oacl  6614  diffifi  7064  djuexb  7222  pitonn  8046  axicn  8061  pnfnre  8199  mnfnre  8200  0mnnnnn0  9412  pfxccatin12lem3  11279  pfxccat3  11281  swrdccat  11282  pfxccat3a  11285  swrdccat3blem  11286  swrdccat3b  11287  nprmi  12661  issubm  13520  issrg  13943  srgfcl  13951  subrngrng  14181  txdis1cn  14967  xmeterval  15124  expcncf  15298  gausslemma2dlem1a  15752  2lgslem4  15797  bj-sucexg  16340