Theorem eqeltrrdi 2231

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 2145	. 2 |- ( ph -> A = B )
3		eqeltrrdi.2	. 2 |- B e. C
4	2, 3	eqeltrdi 2230	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = 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:  eusvnfb  4375  releldm2  6083  mapprc  6546  ixpprc  6613  ixpssmap2g  6621  ixpssmapg  6622  bren  6641  brdomg  6642  mapen  6740  ssenen  6745  fi0  6863  ioof  9757  hashfacen  10582  fsum3  11159  cnrehmeocntop  12765