Theorem eleq12i 2207

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 2206	. 2 |- ( A e. C <-> A e. D )
3		eleq1i.1	. . 3 |- A = B
4	3	eleq1i 2205	. 2 |- ( A e. D <-> B e. D )
5	2, 4	bitri 183	1 |- ( A e. C <-> B e. D )

Syntax hints:   <-> wb 104   = wceq 1331   e. wcel 1480

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-clel 2135

This theorem is referenced by:  3eltr3g  2224  3eltr4g  2225  sbcel12g  3017  ennnfonelem1  11920