Theorem eqeltrrdi 2269

Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Hypotheses

Ref	Expression
eqeltrrdi.1	|- ( ph -> B = A )
eqeltrrdi.2	|- B e. C

Assertion

Ref	Expression
eqeltrrdi	|- ( ph -> A e. C )

Proof of Theorem eqeltrrdi

Step	Hyp	Ref	Expression
1		eqeltrrdi.1	. . 3 |- ( ph -> B = A )
2	1	eqcomd 2183	. 2 |- ( ph -> A = B )
3		eqeltrrdi.2	. 2 |- B e. C
4	2, 3	eqeltrdi 2268	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173

This theorem is referenced by:  eusvnfb  4454  releldm2  6185  mapprc  6651  ixpprc  6718  ixpssmap2g  6726  ixpssmapg  6727  bren  6746  brdomg  6747  mapen  6845  ssenen  6850  fi0  6973  nnnninf2  7124  ioof  9970  hashfacen  10815  fsum3  11394  cnrehmeocntop  14063