Theorem eleqtrrd 1554

Description: Deduction that substitutes equal classes into membership.

Hypotheses

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

Assertion

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

Proof of Theorem eleqtrrd

Step	Hyp	Ref	Expression
1		eleqtrrd.1	. 2 |- (ph -> A e. B)
2		eleqtrrd.2	. . 3 |- (ph -> C = B)
3	2	eqcomd 1483	. 2 |- (ph -> B = C)
4	1, 3	eleqtrd 1553	1 |- (ph -> A e. C)

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

This theorem is referenced by:  tfrlem13 3929  elimdeloprv 4007  omordi 4203  oneo 4218  unblem3 4553  metelcls 7962  imsdval 8313  nvlmcl 8328  spansnid 9481  elspansn4t 9491  rcmob 10653

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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