Theorem eleq12i 2234

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 2233	. 2 |- ( A e. C <-> A e. D )
3		eleq1i.1	. . 3 |- A = B
4	3	eleq1i 2232	. 2 |- ( A e. D <-> B e. D )
5	2, 4	bitri 183	1 |- ( A e. C <-> B e. D )

Syntax hints:   <-> wb 104   = wceq 1343   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158  df-clel 2161

This theorem is referenced by:  3eltr3g  2251  3eltr4g  2252  sbcel12g  3060  ennnfonelem1  12340