Theorem pm11.53 2344

Description: Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 1947 for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
pm11.53	⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥

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

Proof of Theorem pm11.53

Step	Hyp	Ref	Expression
1		19.21v 1942	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓))
2	1	albii 1822	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
3		nfv 1917	. . . 4 ⊢ Ⅎ𝑥𝜓
4	3	nfal 2317	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜓
5	4	19.23 2204	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
6	2, 5	bitri 274	1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782

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  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787

This theorem is referenced by: (None)