Theorem eleq1i 2255

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 2252	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)

Syntax hints:   ↔ wb 105   = wceq 1364   ∈ wcel 2160

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-cleq 2182  df-clel 2185

This theorem is referenced by:  eleq12i  2257  eqeltri  2262  intexrabim  4171  abssexg  4200  abnex  4465  snnex  4466  pwexb  4492  sucexb  4514  omex  4610  iprc  4913  dfse2  5019  fressnfv  5724  fnotovb  5939  f1stres  6184  f2ndres  6185  ottposg  6280  dftpos4  6288  frecabex  6423  oacl  6485  diffifi  6922  djuexb  7073  pitonn  7877  axicn  7892  pnfnre  8029  mnfnre  8030  0mnnnnn0  9238  nprmi  12156  issubm  12924  issrg  13319  srgfcl  13327  subrngrng  13549  txdis1cn  14238  xmeterval  14395  expcncf  14552  bj-sucexg  15135