Theorem pm10.56 37591

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		pm3.45 875	. . 3 ⊢ ((𝜑 → 𝜓) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜒)))
2	1	aleximi 1749	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥(𝜑 ∧ 𝜒) → ∃𝑥(𝜓 ∧ 𝜒)))
3	2	imp 444	1 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥(𝜑 ∧ 𝜒)) → ∃𝑥(𝜓 ∧ 𝜒))

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by: (None)