Theorem eleq12i 2829

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 2828	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)
3		eleq1i.1	. . 3 ⊢ 𝐴 = 𝐵
4	3	eleq1i 2827	. 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷)
5	2, 4	bitri 275	1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2728  df-clel 2811

This theorem is referenced by:  sbcel12  4363  smndex1n0mnd  18837  zclmncvs  25104  gausslemma2dlem4  27336  bnj98  35023  elmpst  35730  elmpps  35767  sbceqbii  36385  cbvsbcvw2  36424  oaordnrex  43537  omnord1ex  43546  oenord1ex  43557  wfaxpow  45238  unirnmapsn  45458  gpgprismgr4cycllem8  48348