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Metamath Proof Explorer

Theorem rexlimddv

Description: Restricted existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016)

Ref Expression
Hypotheses rexlimddv.1 ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
rexlimddv.2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) → 𝜒 )
Assertion rexlimddv ( 𝜑 → 𝜒 )

Proof

Step Hyp Ref Expression
1 rexlimddv.1 ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
2 rexlimddv.2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) → 𝜒 )
3 2 rexlimdvaa ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )
4 1 3 mpd ⊢ ( 𝜑 → 𝜒 )