Theorem eleq12i 2858

Description: Inference from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)

Hypotheses

Ref	Expression
eleq1i.1	⊢ 𝐴 = 𝐵
eleq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
eleq12i	⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)

Proof of Theorem eleq12i

Step	Hyp	Ref	Expression
1		eleq12i.2	. . 3 ⊢ 𝐶 = 𝐷
2	1	eleq2i 2857	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)
3		eleq1i.1	. . 3 ⊢ 𝐴 = 𝐵
4	3	eleq1i 2856	. 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷)
5	2, 4	bitri 278	1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2146

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2757  df-clel 2840

This theorem is used by:  sbcel12  4376  smndex1n0mnd  19013  zclmncvs  25360  gausslemma2dlem4  27586  bnj98  35322  elmpst  36067  elmpps  36104  sbceqbii  36762  cbvsbcvw2  36801  oaordnrex  44082  omnord1ex  44091  oenord1ex  44102  wfaxpow  45766  unirnmapsn  45990  gpgprismgr4cycllem8  48927  isprmrng  49160