Theorem rspceb2dv

Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by Zhi Wang, 28-Sep-2024)

	Ref	Expression
Hypotheses	rspceb2dv.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝜓 → 𝜒 ) )
	rspceb2dv.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝐴 ∈ 𝐵 )
	rspceb2dv.3	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
	rspceb2dv.4	⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜃 ) )
Assertion	rspceb2dv	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 𝜓 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		rspceb2dv.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → ( 𝜓 → 𝜒 ) )
2		rspceb2dv.2	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝐴 ∈ 𝐵 )
3		rspceb2dv.3	⊢ ( ( 𝜑 ∧ 𝜒 ) → 𝜃 )
4		rspceb2dv.4	⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜃 ) )
5	1	rexlimdva	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 𝜓 → 𝜒 ) )
6	4	rspcev	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜃 ) → ∃ 𝑥 ∈ 𝐵 𝜓 )
7	2 3 6	syl2anc	⊢ ( ( 𝜑 ∧ 𝜒 ) → ∃ 𝑥 ∈ 𝐵 𝜓 )
8	7	ex	⊢ ( 𝜑 → ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜓 ) )
9	5 8	impbid	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 𝜓 ↔ 𝜒 ) )