Theorem eqneltrd 2446

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

Hypotheses

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

Assertion

Ref	Expression
eqneltrd	⊢ (φ → ¬ A ∈ C)

Proof of Theorem eqneltrd

Step	Hyp	Ref	Expression
1		eqneltrd.2	. 2 ⊢ (φ → ¬ B ∈ C)
2		eqneltrd.1	. . 3 ⊢ (φ → A = B)
3	2	eleq1d 2419	. 2 ⊢ (φ → (A ∈ C ↔ B ∈ C))
4	1, 3	mtbird 292	1 ⊢ (φ → ¬ A ∈ C)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349

This theorem is referenced by: (None)