Theorem pm11.53 1893

Description: Theorem *11.53 in [WhiteheadRussell] p. 164. (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 1890	. . 3 ⊢ (∀y(φ → ψ) ↔ (φ → ∀yψ))
2	1	albii 1566	. 2 ⊢ (∀x∀y(φ → ψ) ↔ ∀x(φ → ∀yψ))
3		nfv 1619	. . . 4 ⊢ Ⅎxψ
4	3	nfal 1842	. . 3 ⊢ Ⅎx∀yψ
5	4	19.23 1801	. 2 ⊢ (∀x(φ → ∀yψ) ↔ (∃xφ → ∀yψ))
6	2, 5	bitri 240	1 ⊢ (∀x∀y(φ → ψ) ↔ (∃xφ → ∀yψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)