Theorem pm10.542 41986

Description: Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm10.542	⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) ↔ (𝜒 → ∀𝑥(𝜑 → 𝜓)))

Distinct variable group:   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem pm10.542

Step	Hyp	Ref	Expression
1		bi2.04 389	. . 3 ⊢ ((𝜑 → (𝜒 → 𝜓)) ↔ (𝜒 → (𝜑 → 𝜓)))
2	1	albii 1822	. 2 ⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) ↔ ∀𝑥(𝜒 → (𝜑 → 𝜓)))
3		19.21v 1942	. 2 ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) ↔ (𝜒 → ∀𝑥(𝜑 → 𝜓)))
4	2, 3	bitri 274	1 ⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) ↔ (𝜒 → ∀𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by: (None)