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 ) )