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Metamath Proof Explorer

Theorem rspcdv

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 17-Feb-2007) (Revised by Mario Carneiro, 4-Jan-2017)

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

Proof

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