Theorem rspcdv2

Description: Restricted specialization, using implicit substitution. (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypotheses	rspcdv2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
	rspcdv2.2	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
	rspcdv2.3	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 𝜓 )
Assertion	rspcdv2	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		rspcdv2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
2		rspcdv2.2	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
3		rspcdv2.3	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 𝜓 )
4	2 1	rspcdv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 → 𝜒 ) )
5	3 4	mpd	⊢ ( 𝜑 → 𝜒 )