Theorem eleqtrd 1165

Description: Deduction that substitutes equal classes into membership.

Hypotheses

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

Assertion

Ref	Expression
eleqtrd	⊢ (φ → A ∈ C)

Proof of Theorem eleqtrd

Step	Hyp	Ref	Expression
1		eleqtrd.1	. 2 ⊢ (φ → A ∈ B)
2		eleqtrd.2	. . 3 ⊢ (φ → B = C)
3	2	eleq2d 1156	. 2 ⊢ (φ → (A ∈ B ↔ A ∈ C))
4	1, 3	mpbid 170	1 ⊢ (φ → A ∈ C)

Syntax hints:   → wi 2   = wceq 1091   ∈ 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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