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 1398   ∈ 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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  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  4248  abssexg  4278  abnex  4550  snnex  4551  pwexb  4577  sucexb  4601  omex  4697  iprc  5007  dfse2  5116  fressnfv  5849  fnotovb  6074  f1stres  6331  f2ndres  6332  ottposg  6464  dftpos4  6472  frecabex  6607  oacl  6671  diffifi  7126  djuexb  7286  pitonn  8111  axicn  8126  pnfnre  8263  mnfnre  8264  0mnnnnn0  9476  pfxccatin12lem3  11362  pfxccat3  11364  swrdccat  11365  pfxccat3a  11368  swrdccat3blem  11369  swrdccat3b  11370  nprmi  12759  issubm  13618  issrg  14042  srgfcl  14050  subrngrng  14280  txdis1cn  15072  xmeterval  15229  expcncf  15403  gausslemma2dlem1a  15860  2lgslem4  15905  clwwlknonex2  16363  bj-sucexg  16621