Theorem syl5eleqr 1555

Description: A membership and equality inference.

Hypotheses

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

Assertion

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

Proof of Theorem syl5eleqr

Step	Hyp	Ref	Expression
1		syl5eleqr.1	. . 3 |- (ph -> B = A)
2	1	eqcomd 1480	. 2 |- (ph -> A = B)
3		syl5eleqr.2	. 2 |- C e. A
4	2, 3	syl5eleq 1554	1 |- (ph -> C e. B)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958

This theorem is referenced by:  opprc1b 2796  oen0 4213  inf0 4606  r1ord 4655  ruclem37 7546  vcoprne 8198

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-cleq 1469  df-clel 1472

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