Theorem pm10.56 40708

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

Assertion

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

Proof of Theorem pm10.56

Step	Hyp	Ref	Expression
1		pm3.45 623	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)))
2	1	aleximi 1831	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥(𝜑 ∧ 𝜒) → ∃𝑥(𝜓 ∧ 𝜒)))
3	2	imp 409	1 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1534  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780

This theorem is referenced by: (None)