Theorem eleq12i 2825

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 2824	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)
3		eleq1i.1	. . 3 ⊢ 𝐴 = 𝐵
4	3	eleq1i 2823	. 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷)
5	2, 4	bitri 275	1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)

Syntax hints:   ↔ wb 205   = wceq 1540   ∈ wcel 2105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-cleq 2723  df-clel 2809

This theorem is referenced by:  sbcel12  4408  smndex1n0mnd  18835  zclmncvs  24996  gausslemma2dlem4  27216  bnj98  34343  elmpst  34992  elmpps  35029  oaordnrex  42510  omnord1ex  42519  oenord1ex  42530  unirnmapsn  44374