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Metamath Proof Explorer

Theorem raldifsni

Description: Rearrangement of a property of a singleton difference. (Contributed by Stefan O'Rear, 27-Feb-2015)

Ref Expression
Assertion raldifsni ( ∀ 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) ¬ 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝑥 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 eldifsn ⊢ ( 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ) )
2 1 imbi1i ⊢ ( ( 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) → ¬ 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ) → ¬ 𝜑 ) )
3 impexp ⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ) → ¬ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑥 ≠ 𝐵 → ¬ 𝜑 ) ) )
4 df-ne ⊢ ( 𝑥 ≠ 𝐵 ↔ ¬ 𝑥 = 𝐵 )
5 4 imbi1i ⊢ ( ( 𝑥 ≠ 𝐵 → ¬ 𝜑 ) ↔ ( ¬ 𝑥 = 𝐵 → ¬ 𝜑 ) )
6 con34b ⊢ ( ( 𝜑 → 𝑥 = 𝐵 ) ↔ ( ¬ 𝑥 = 𝐵 → ¬ 𝜑 ) )
7 5 6 bitr4i ⊢ ( ( 𝑥 ≠ 𝐵 → ¬ 𝜑 ) ↔ ( 𝜑 → 𝑥 = 𝐵 ) )
8 7 imbi2i ⊢ ( ( 𝑥 ∈ 𝐴 → ( 𝑥 ≠ 𝐵 → ¬ 𝜑 ) ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝑥 = 𝐵 ) ) )
9 2 3 8 3bitri ⊢ ( ( 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) → ¬ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝑥 = 𝐵 ) ) )
10 9 ralbii2 ⊢ ( ∀ 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) ¬ 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝑥 = 𝐵 ) )