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Metamath Proof Explorer

Theorem neleq12d

Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016) (Proof shortened by Wolf Lammen, 25-Nov-2019)

Ref Expression
Hypotheses neleq12d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
neleq12d.2 ⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion neleq12d ( 𝜑 → ( 𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷 ) )

Proof

Step Hyp Ref Expression
1 neleq12d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 neleq12d.2 ⊢ ( 𝜑 → 𝐶 = 𝐷 )
3 1 2 eleq12d ⊢ ( 𝜑 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷 ) )
4 3 notbid ⊢ ( 𝜑 → ( ¬ 𝐴 ∈ 𝐶 ↔ ¬ 𝐵 ∈ 𝐷 ) )
5 df-nel ⊢ ( 𝐴 ∉ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶 )
6 df-nel ⊢ ( 𝐵 ∉ 𝐷 ↔ ¬ 𝐵 ∈ 𝐷 )
7 4 5 6 3bitr4g ⊢ ( 𝜑 → ( 𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷 ) )