Theorem spcdv

Description: Rule of specialization, using implicit substitution. Analogous to rspcdv . (Contributed by David Moews, 1-May-2017)

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

Proof

Step	Hyp	Ref	Expression
1		spcimdv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		spcdv.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3	2	biimpd	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 → 𝜒 ) )
4	1 3	spcimdv	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) )