Theorem eleqtrd 1550

Description: Deduction that substitutes equal classes into membership.

Hypotheses

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

Assertion

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

Proof of Theorem eleqtrd

Step	Hyp	Ref	Expression
1		eleqtrd.1	. 2 |- (ph -> A e. B)
2		eleqtrd.2	. . 3 |- (ph -> B = C)
3	2	eleq2d 1541	. 2 |- (ph -> (A e. B <-> A e. C))
4	1, 3	mpbid 195	1 |- (ph -> A e. C)

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

This theorem is referenced by:  eleqtrrd 1551  syl5eleq 1554  syl6eleq 1558  rankxplim3 4714  fsum0split 7021  cnpco 7769  lpbl 7880  nvlmle 8333  homib 10724

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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