Theorem eleq1a 2310

Description: A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)

Assertion

Ref	Expression
eleq1a	|- ( A e. B -> ( C = A -> C e. B ) )

Proof of Theorem eleq1a

Step	Hyp	Ref	Expression
1		eleq1 2301	. 2 |- ( C = A -> ( C e. B <-> A e. B ) )
2	1	biimprcd 160	1 |- ( A e. B -> ( C = A -> C e. B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234

This theorem is used by:  elex22  2837  elex2  2838  reu6  3015  disjne  3578  ssimaex  5767  fnex  5940  f1ocnv2d  6297  f1o3d  6301  mpoexw  6453  tfrlem8  6593  eroprf  6906  ac6sfi  7206  recclnq  7765  prnmaddl  7863  mpomulf  8322  renegcl  8594  nn0ind-raph  9774  iccid  10344  4sqlem1  13196  4sqlem4  13200  4sqlem11  13209  lssvneln0  14819  lss1d  14829  lspsn  14862  rnglidlmmgm  14942  opnneiid  15390  metrest  15732  coseq0negpitopi  16063  ppiublem1  16291  bj-nn0suc  17195  bj-inf2vnlem2  17202  bj-nn0sucALT  17209