Theorem eleqtrri 2305

Description: Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
eleqtrr.1	|- A e. B
eleqtrr.2	|- C = B

Assertion

Ref	Expression
eleqtrri	|- A e. C

Proof of Theorem eleqtrri

Step	Hyp	Ref	Expression
1		eleqtrr.1	. 2 |- A e. B
2		eleqtrr.2	. . 3 |- C = B
3	2	eqcomi 2233	. 2 |- B = C
4	1, 3	eleqtri 2304	1 |- A e. C

Syntax hints:   = wceq 1395   e. wcel 2200

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225

This theorem is referenced by:  3eltr4i  2311  undifexmid  4277  opi1  4318  opi2  4319  ordpwsucexmid  4662  peano1  4686  acexmidlemcase  5996  acexmidlem2  5998  0lt2o  6587  1lt2o  6588  0elixp  6876  ac6sfi  7060  ctssdccl  7278  exmidomni  7309  exmidonfinlem  7371  exmidfodomrlemr  7380  exmidfodomrlemrALT  7381  exmidaclem  7390  pw1ne3  7415  3nelsucpw1  7419  1lt2pi  7527  prarloclemarch2  7606  prarloclemlt  7680  prarloclemcalc  7689  suplocexprlemdisj  7907  suplocexprlemub  7910  pnfxr  8199  mnfxr  8203  0bits  12470  fnpr2ob  13373  dveflem  15400