Theorem syl6eqelr 1549

Description: A membership and equality inference.

Hypotheses

Ref	Expression
syl6eqelr.1	|- (ph -> B = A)
syl6eqelr.2	|- B e. C

Assertion

Ref	Expression
syl6eqelr	|- (ph -> A e. C)

Proof of Theorem syl6eqelr

Step	Hyp	Ref	Expression
1		syl6eqelr.1	. . 3 |- (ph -> B = A)
2	1	eqcomd 1472	. 2 |- (ph -> A = B)
3		syl6eqelr.2	. 2 |- B e. C
4	2, 3	syl6eqel 1548	1 |- (ph -> A e. C)

Syntax hints:   -> wi 3   = wceq 953   e. wcel 955

This theorem is referenced by:  mapprc 4310  breng 4357  brdomg 4358  pw2en 4426  oncardon 4792  oncardid 4793  cardcf 4883  cfeq0 4886  nnsub 5903  ioof 6333  sqrcl 6630  hsupclt 9222

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-17 968  ax-4 970  ax-5o 972  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-cleq 1462  df-clel 1465

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