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Metamath Proof Explorer

Theorem rspcime

Description: Prove a restricted existential. (Contributed by Rohan Ridenour, 3-Aug-2023)

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

Proof

Step Hyp Ref Expression
1 rspcime.1 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝜓 )
2 rspcime.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
3 simpl ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝜑 )
4 1 3 2thd ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜑 ) )
5 id ⊢ ( 𝜑 → 𝜑 )
6 2 4 5 rspcedvd ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐵 𝜓 )