Theorem rexsng

Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012) Avoid ax-10 , ax-12 . (Revised by GG, 30-Sep-2024)

	Ref	Expression
Hypothesis	ralsng.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rexsng	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ralsng.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2	1	notbid	⊢ ( 𝑥 = 𝐴 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
3	2	ralsng	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ∈ { 𝐴 } ¬ 𝜑 ↔ ¬ 𝜓 ) )
4		dfrex2	⊢ ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ ¬ ∀ 𝑥 ∈ { 𝐴 } ¬ 𝜑 )
5		bicom1	⊢ ( ( ∀ 𝑥 ∈ { 𝐴 } ¬ 𝜑 ↔ ¬ 𝜓 ) → ( ¬ 𝜓 ↔ ∀ 𝑥 ∈ { 𝐴 } ¬ 𝜑 ) )
6	5	con1bid	⊢ ( ( ∀ 𝑥 ∈ { 𝐴 } ¬ 𝜑 ↔ ¬ 𝜓 ) → ( ¬ ∀ 𝑥 ∈ { 𝐴 } ¬ 𝜑 ↔ 𝜓 ) )
7	4 6	bitrid	⊢ ( ( ∀ 𝑥 ∈ { 𝐴 } ¬ 𝜑 ↔ ¬ 𝜓 ) → ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )
8	3 7	syl	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )