Theorem eleqtrd 1553

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 1544	. 2 |- (ph -> (A e. B <-> A e. C))
4	1, 3	mpbid 195	1 |- (ph -> A e. C)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960

This theorem is referenced by:  eleqtrrd 1554  syl5eleq 1557  syl6eleq 1561  rankxplim3 4724  fsum0split 7021  cnpco 7766  lpbl 7877  nvlmle 8329  homib 10695

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-cleq 1472  df-clel 1475

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