Theorem pm10.56

Description: Theorem *10.56 in WhiteheadRussell p. 156. (Contributed by Andrew Salmon, 24-May-2011)

	Ref	Expression
Assertion	pm10.56	⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜒 ) ) → ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		pm3.45	⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 ∧ 𝜒 ) → ( 𝜓 ∧ 𝜒 ) ) )
2	1	aleximi	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 ( 𝜑 ∧ 𝜒 ) → ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) ) )
3	2	imp	⊢ ( ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 ( 𝜑 ∧ 𝜒 ) ) → ∃ 𝑥 ( 𝜓 ∧ 𝜒 ) )