Theorem syl6eleq 2130

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

Hypotheses

Ref	Expression
syl6eleq.1	|- ( ph -> A e. B )
syl6eleq.2	|- B = C

Assertion

Ref	Expression
syl6eleq	|- ( ph -> A e. C )

Proof of Theorem syl6eleq

Step	Hyp	Ref	Expression
1		syl6eleq.1	. 2 |- ( ph -> A e. B )
2		syl6eleq.2	. . 3 |- B = C
3	2	a1i 9	. 2 |- ( ph -> B = C )
4	1, 3	eleqtrd 2116	1 |- ( ph -> A e. C )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by:  syl6eleqr  2131  prid2g  3475  caucvgprprlem2  6808  gt0srpr  6833  eluzel2  8478  fseq1p1m1  8956  fznn0sub2  8985  nn0split  8994  exple1  9310  clim2iser  9857  clim2iser2  9858  iiserex  9859  iisermulc2  9860  iserile  9862  iserige0  9863  climub  9864  climserile  9865  serif0  9871  ialgrp1  9885