Theorem pm10.542 37589

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		bi2.04 375	. . 3 ⊢ ((𝜑 → (𝜒 → 𝜓)) ↔ (𝜒 → (𝜑 → 𝜓)))
2	1	albii 1737	. 2 ⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) ↔ ∀𝑥(𝜒 → (𝜑 → 𝜓)))
3		19.21v 1855	. 2 ⊢ (∀𝑥(𝜒 → (𝜑 → 𝜓)) ↔ (𝜒 → ∀𝑥(𝜑 → 𝜓)))
4	2, 3	bitri 263	1 ⊢ (∀𝑥(𝜑 → (𝜒 → 𝜓)) ↔ (𝜒 → ∀𝑥(𝜑 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827

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

This theorem is referenced by: (None)