Theorem rspcedvdw

Description: Version of rspcedvd where the implicit substitution hypothesis does not have an antecedent, which also avoids a disjoint variable condition on ph , x . (Contributed by SN, 20-Aug-2024)

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

Proof

Step	Hyp	Ref	Expression
1		rspcedvdw.s	⊢ ( 𝑥 = 𝐴 → ( 𝜓 ↔ 𝜒 ) )
2		rspcedvdw.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
3		rspcedvdw.2	⊢ ( 𝜑 → 𝜒 )
4	1	rspcev	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜒 ) → ∃ 𝑥 ∈ 𝐵 𝜓 )
5	2 3 4	syl2anc	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 )