Theorem eleqtrrd 1166

Description: Deduction that substitutes equal classes into membership.

Hypotheses

Ref	Expression
eleqtrrd.1	⊢ (φ → A ∈ B)
eleqtrrd.2	⊢ (φ → C = B)

Assertion

Ref	Expression
eleqtrrd	⊢ (φ → A ∈ C)

Proof of Theorem eleqtrrd

Step	Hyp	Ref	Expression
1		eleqtrrd.1	. 2 ⊢ (φ → A ∈ B)
2		eleqtrrd.2	. . 3 ⊢ (φ → C = B)
3	2	cleqcomd 1106	. 2 ⊢ (φ → B = C)
4	1, 3	eleqtrd 1165	1 ⊢ (φ → A ∈ C)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  tfrlem13 2961  omordi 3164  unblem3 3433  spansnid 5468  elspansn4t 5478

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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