Theorem eleqtrrd 1551

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 1480	. 2 |- (ph -> B = C)
4	1, 3	eleqtrd 1550	1 |- (ph -> A e. C)

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

This theorem is referenced by:  tfrlem13 3923  elimdeloprv 4001  omordi 4197  oneo 4212  unblem3 4542  metelcls 7965  imsdval 8317  nvlmcl 8332  spwpr3OLD 8662  spansnid 9486  elspansn4t 9496  rcmob 10682

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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