Theorem rspcdva

Description: Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020)

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

Proof

Step	Hyp	Ref	Expression
1		rspcdva.1	⊢ ( 𝑥 = 𝐶 → ( 𝜓 ↔ 𝜒 ) )
2		rspcdva.2	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )
3		rspcdva.3	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
4	1	rspcv	⊢ ( 𝐶 ∈ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )
5	3 2 4	sylc	⊢ ( 𝜑 → 𝜒 )