Theorem eleq1i 2230

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 2227	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)

Syntax hints:   ↔ wb 104   = wceq 1342   ∈ wcel 2135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-cleq 2157  df-clel 2160

This theorem is referenced by:  eleq12i  2232  eqeltri  2237  intexrabim  4126  abssexg  4155  abnex  4419  snnex  4420  pwexb  4446  sucexb  4468  omex  4564  iprc  4866  dfse2  4971  fressnfv  5666  fnotovb  5876  f1stres  6119  f2ndres  6120  ottposg  6214  dftpos4  6222  frecabex  6357  oacl  6419  diffifi  6851  djuexb  7000  pitonn  7780  axicn  7795  pnfnre  7931  mnfnre  7932  0mnnnnn0  9137  nprmi  12035  txdis1cn  12819  xmeterval  12976  expcncf  13133  bj-sucexg  13639