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  5841  fnotovb  6064  f1stres  6322  f2ndres  6323  ottposg  6421  dftpos4  6429  frecabex  6564  oacl  6628  diffifi  7083  djuexb  7243  pitonn  8068  axicn  8083  pnfnre  8221  mnfnre  8222  0mnnnnn0  9434  pfxccatin12lem3  11317  pfxccat3  11319  swrdccat  11320  pfxccat3a  11323  swrdccat3blem  11324  swrdccat3b  11325  nprmi  12701  issubm  13560  issrg  13984  srgfcl  13992  subrngrng  14222  txdis1cn  15008  xmeterval  15165  expcncf  15339  gausslemma2dlem1a  15793  2lgslem4  15838  clwwlknonex2  16296  bj-sucexg  16543