Theorem eleq12i 2146

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 e. C <-> B e. D )

Proof of Theorem eleq12i

Step	Hyp	Ref	Expression
1		eleq12i.2	. . 3 |- C = D
2	1	eleq2i 2145	. 2 |- ( A e. C <-> A e. D )
3		eleq1i.1	. . 3 |- A = B
4	3	eleq1i 2144	. 2 |- ( A e. D <-> B e. D )
5	2, 4	bitri 182	1 |- ( A e. C <-> B e. D )

Syntax hints:   <-> wb 103   = wceq 1284   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077

This theorem is referenced by:  3eltr3g  2163  3eltr4g  2164  sbcel12g  2921