Theorem eleqtrr 1162

Description: Substitution of equal classes into membership relation.

Hypotheses

Ref	Expression
eleqtrr.1	⊢ A ∈ B
eleqtrr.2	⊢ C = B

Assertion

Ref	Expression
eleqtrr	⊢ A ∈ C

Proof of Theorem eleqtrr

Step	Hyp	Ref	Expression
1		eleqtrr.1	. 2 ⊢ A ∈ B
2		eleqtrr.2	. . 3 ⊢ C = B
3	2	cleqcomi 1105	. 2 ⊢ B = C
4	1, 3	eleqtr 1161	1 ⊢ A ∈ C

Syntax hints:   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  opi1 1895  opi2 1896  pw2en 3348  tz9.13 3507  rankid 3516  rankpw 3528  1lt2pi 3826  indpi 3828  1nn 4432  projlem8 5200

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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