Theorem eleq12i 2822

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 2821	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)
3		eleq1i.1	. . 3 ⊢ 𝐴 = 𝐵
4	3	eleq1i 2820	. 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷)
5	2, 4	bitri 274	1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)

Syntax hints:   ↔ wb 205   = wceq 1533   ∈ wcel 2098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2720  df-clel 2806

This theorem is referenced by:  sbcel12  4412  smndex1n0mnd  18871  zclmncvs  25096  gausslemma2dlem4  27322  bnj98  34531  elmpst  35179  elmpps  35216  oaordnrex  42755  omnord1ex  42764  oenord1ex  42775  unirnmapsn  44617