Theorem rspcebdv

Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022)

	Ref	Expression
Hypotheses	rspcdv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	rspcdv.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
	rspcebdv.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑥 = 𝐴 )
Assertion	rspcebdv	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 𝜓 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rspcdv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		rspcdv.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3		rspcebdv.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑥 = 𝐴 )
4	3 2	syldan	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜓 ↔ 𝜒 ) )
5	4	biimpd	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜓 → 𝜒 ) )
6	5	expcom	⊢ ( 𝜓 → ( 𝜑 → ( 𝜓 → 𝜒 ) ) )
7	6	pm2.43b	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
8	7	rexlimdvw	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 𝜓 → 𝜒 ) )
9	1 2	rspcedv	⊢ ( 𝜑 → ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜓 ) )
10	8 9	impbid	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 𝜓 ↔ 𝜒 ) )