Theorem eleq12i 2272

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 2271	. 2 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)
3		eleq1i.1	. . 3 ⊢ 𝐴 = 𝐵
4	3	eleq1i 2270	. 2 ⊢ (𝐴 ∈ 𝐷 ↔ 𝐵 ∈ 𝐷)
5	2, 4	bitri 184	1 ⊢ (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷)

Syntax hints:   ↔ wb 105   = wceq 1372   ∈ wcel 2175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-cleq 2197  df-clel 2200

This theorem is referenced by:  3eltr3g  2289  3eltr4g  2290  sbcel12g  3107  ennnfonelem1  12749  gausslemma2dlem4  15512