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Metamath Proof Explorer

Theorem rspcimdv

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017)

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

Proof

Step Hyp Ref Expression
1 rspcimdv.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2 rspcimdv.2 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 → 𝜒 ) )
3 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐵 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜓 ) )
4 simpr ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝑥 = 𝐴 )
5 4 eleq1d ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
6 5 biimprd ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐵 ) )
7 6 2 imim12d ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( ( 𝑥 ∈ 𝐵 → 𝜓 ) → ( 𝐴 ∈ 𝐵 → 𝜒 ) ) )
8 1 7 spcimdv ⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜓 ) → ( 𝐴 ∈ 𝐵 → 𝜒 ) ) )
9 1 8 mpid ⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 ∈ 𝐵 → 𝜓 ) → 𝜒 ) )
10 3 9 biimtrid ⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 𝜓 → 𝜒 ) )