Theorem eleqtrd 1165

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 1156	. 2 |- (ph -> (A e. B <-> A e. C))
4	1, 3	mpbid 170	1 |- (ph -> A e. C)

Syntax hints:   -> wi 2   = wceq 1091   e. wcel 1092

This theorem is referenced by:  eleqtrrd 1166

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099

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