Theorem eqneltrrd 2329

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
eqneltrrd.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqneltrrd.2	⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)

Assertion

Ref	Expression
eqneltrrd	⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)

Proof of Theorem eqneltrrd

Step	Hyp	Ref	Expression
1		eqneltrrd.2	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)
2		eqneltrrd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	eleq1d 2301	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶))
4	1, 3	mtbid 679	1 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1398   ∈ wcel 2203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-cleq 2225  df-clel 2228

This theorem is referenced by:  exmidapne  7570  ctinf  13170  lssvancl2  14503