Theorem eleq1i 2148

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 2145	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
3	1, 2	ax-mp 7	1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶)

Syntax hints:   ↔ wb 103   = wceq 1285   ∈ wcel 1434

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-cleq 2076  df-clel 2079

This theorem is referenced by:  eleq12i  2150  eqeltri  2155  intexrabim  3954  abssexg  3981  snnex  4235  pwexb  4260  sucexb  4277  omex  4371  iprc  4659  dfse2  4760  fressnfv  5426  fnotovb  5627  f1stres  5865  f2ndres  5866  ottposg  5952  dftpos4  5960  frecabex  6095  oacl  6153  diffifi  6540  pitonn  7288  axicn  7303  pnfnre  7432  mnfnre  7433  0mnnnnn0  8597  nprmi  10886  bj-sucexg  11156