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Metamath Proof Explorer

Theorem bnj1023

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1023.1 ⊢ ∃ x φ → ψ
bnj1023.2 ⊢ ψ → χ
Assertion bnj1023 ⊢ ∃ x φ → χ

Proof

Step Hyp Ref Expression
1 bnj1023.1 ⊢ ∃ x φ → ψ
2 bnj1023.2 ⊢ ψ → χ
3 2 a1i ⊢ φ → ψ → ψ → χ
4 3 ax-gen ⊢ ∀ x φ → ψ → ψ → χ
5 exintr ⊢ ∀ x φ → ψ → ψ → χ → ∃ x φ → ψ → ∃ x φ → ψ ∧ ψ → χ
6 4 1 5 mp2 ⊢ ∃ x φ → ψ ∧ ψ → χ
7 pm3.33 ⊢ φ → ψ ∧ ψ → χ → φ → χ
8 6 7 bnj101 ⊢ ∃ x φ → χ