Theorem rexdifsn

Description: Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015)

	Ref	Expression
Assertion	rexdifsn	⊢ ( ∃ 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 ( 𝑥 ≠ 𝐵 ∧ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		eldifsn	⊢ ( 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ) )
2	1	anbi1i	⊢ ( ( 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) ∧ 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ) ∧ 𝜑 ) )
3		anass	⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ≠ 𝐵 ) ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ≠ 𝐵 ∧ 𝜑 ) ) )
4	2 3	bitri	⊢ ( ( 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) ∧ 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ≠ 𝐵 ∧ 𝜑 ) ) )
5	4	rexbii2	⊢ ( ∃ 𝑥 ∈ ( 𝐴 ∖ { 𝐵 } ) 𝜑 ↔ ∃ 𝑥 ∈ 𝐴 ( 𝑥 ≠ 𝐵 ∧ 𝜑 ) )