Theorem eleq1i 2203

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 2200	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)

Syntax hints:   ↔ wb 104   = wceq 1331   ∈ wcel 1480

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133

This theorem is referenced by:  eleq12i  2205  eqeltri  2210  intexrabim  4073  abssexg  4101  abnex  4363  snnex  4364  pwexb  4390  sucexb  4408  omex  4502  iprc  4802  dfse2  4907  fressnfv  5600  fnotovb  5807  f1stres  6050  f2ndres  6051  ottposg  6145  dftpos4  6153  frecabex  6288  oacl  6349  diffifi  6781  djuexb  6922  pitonn  7649  axicn  7664  pnfnre  7800  mnfnre  7801  0mnnnnn0  9002  nprmi  11794  txdis1cn  12436  xmeterval  12593  expcncf  12750  bj-sucexg  13109