Theorem pm10.12 45101

Description: Theorem *10.12 in [WhiteheadRussell] p. 146. In *10, this is treated as an axiom, and the proofs in *10 are based on this theorem. (Contributed by Andrew Salmon, 17-Jun-2011.)

Assertion

Ref	Expression
pm10.12	⊢ (∀𝑥(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem pm10.12

Step	Hyp	Ref	Expression
1		19.32v 1973	. 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
2	1	biimpi 219	1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (𝜑 ∨ ∀𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813

This theorem is used by:  pm11.12  45118