Theorem syl5eleqr 2440

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

Hypotheses

Ref	Expression
syl5eleqr.1	|- A e. B
syl5eleqr.2	|- (ph -> C = B)

Assertion

Ref	Expression
syl5eleqr	|- (ph -> A e. C)

Proof of Theorem syl5eleqr

Step	Hyp	Ref	Expression
1		syl5eleqr.1	. 2 |- A e. B
2		syl5eleqr.2	. . 3 |- (ph -> C = B)
3	2	eqcomd 2358	. 2 |- (ph -> B = C)
4	1, 3	syl5eleq 2439	1 |- (ph -> A e. C)

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349

This theorem is referenced by:  rabsnt  3798  enadj  6061  enprmaplem5  6081  ce0  6191  te0c  6238