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  4241  abssexg  4270  abnex  4542  snnex  4543  pwexb  4569  sucexb  4593  omex  4689  iprc  4999  dfse2  5107  fressnfv  5836  fnotovb  6059  f1stres  6317  f2ndres  6318  ottposg  6416  dftpos4  6424  frecabex  6559  oacl  6623  diffifi  7076  djuexb  7234  pitonn  8058  axicn  8073  pnfnre  8211  mnfnre  8212  0mnnnnn0  9424  pfxccatin12lem3  11303  pfxccat3  11305  swrdccat  11306  pfxccat3a  11309  swrdccat3blem  11310  swrdccat3b  11311  nprmi  12686  issubm  13545  issrg  13968  srgfcl  13976  subrngrng  14206  txdis1cn  14992  xmeterval  15149  expcncf  15323  gausslemma2dlem1a  15777  2lgslem4  15822  clwwlknonex2  16234  bj-sucexg  16453