Theorem eqeltrrdi 2232

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 2146	. 2 |- ( ph -> A = B )
3		eqeltrrdi.2	. 2 |- B e. C
4	2, 3	eqeltrdi 2231	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-cleq 2133  df-clel 2136

This theorem is referenced by:  eusvnfb  4383  releldm2  6091  mapprc  6554  ixpprc  6621  ixpssmap2g  6629  ixpssmapg  6630  bren  6649  brdomg  6650  mapen  6748  ssenen  6753  fi0  6871  ioof  9784  hashfacen  10611  fsum3  11188  cnrehmeocntop  12801