Theorem ceqsexv

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995)

	Ref	Expression
Hypotheses	ceqsexv.1	⊢ 𝐴 ∈ V
	ceqsexv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ceqsexv	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		ceqsexv.1	⊢ 𝐴 ∈ V
2		ceqsexv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		nfv	⊢ Ⅎ 𝑥 𝜓
4	3 1 2	ceqsex	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ 𝜓 )