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Metamath Proof Explorer

Theorem rspcedvd

Description: Restricted existential specialization, using implicit substitution. Variant of rspcedv . (Contributed by AV, 27-Nov-2019)

Ref Expression
Hypotheses rspcedvd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
rspcedvd.2 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
rspcedvd.3 ⊢ ( 𝜑 → 𝜒 )
Assertion rspcedvd ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 )

Proof

Step Hyp Ref Expression
1 rspcedvd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2 rspcedvd.2 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3 rspcedvd.3 ⊢ ( 𝜑 → 𝜒 )
4 1 2 rspcedv ⊢ ( 𝜑 → ( 𝜒 → ∃ 𝑥 ∈ 𝐵 𝜓 ) )
5 3 4 mpd ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 )