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Metamath Proof Explorer

Theorem neleqtrd

Description: If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017)

Ref Expression
Hypotheses neleqtrd.1 ⊢ φ → ¬ C ∈ A
neleqtrd.2 ⊢ φ → A = B
Assertion neleqtrd ⊢ φ → ¬ C ∈ B

Proof

Step Hyp Ref Expression
1 neleqtrd.1 ⊢ φ → ¬ C ∈ A
2 neleqtrd.2 ⊢ φ → A = B
3 2 eleq2d ⊢ φ → C ∈ A ↔ C ∈ B
4 1 3 mtbid ⊢ φ → ¬ C ∈ B