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Metamath Proof Explorer

Theorem exlimddv

Description: Existential elimination rule of natural deduction (Rule C, explained in exlimiv ). (Contributed by Mario Carneiro, 15-Jun-2016)

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

Proof

Step Hyp Ref Expression
1 exlimddv.1 ⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
2 exlimddv.2 ⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝜒 )
3 2 ex ⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4 3 exlimdv ⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → 𝜒 ) )
5 1 4 mpd ⊢ ( 𝜑 → 𝜒 )