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Metamath Proof Explorer

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 )