Theorem rexsn

Description: Convert an existential quantification restricted to a singleton to a substitution. (Contributed by Jeff Madsen, 5-Jan-2011)

	Ref	Expression
Hypotheses	ralsn.1	⊢ 𝐴 ∈ V
	ralsn.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rexsn	⊢ ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ralsn.1	⊢ 𝐴 ∈ V
2		ralsn.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3	2	rexsng	⊢ ( 𝐴 ∈ V → ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )
4	1 3	ax-mp	⊢ ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 )