Theorem ceqsrexv2

Description: Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017)

	Ref	Expression
Hypothesis	ceqsrexv2.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ceqsrexv2	⊢ ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ceqsrexv2.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2	1	ceqsrexbv	⊢ ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) )