This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem eleqtrrd

Description: Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004)

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

Proof

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