Theorem 19.12vv 1898

Description: Special case of 19.12 1847 where its converse holds. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
19.12vv	⊢ (∃x∀y(φ → ψ) ↔ ∀y∃x(φ → ψ))

Distinct variable groups:   ψ,x   φ,y

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

Proof of Theorem 19.12vv

Step	Hyp	Ref	Expression
1		19.21v 1890	. . 3 ⊢ (∀y(φ → ψ) ↔ (φ → ∀yψ))
2	1	exbii 1582	. 2 ⊢ (∃x∀y(φ → ψ) ↔ ∃x(φ → ∀yψ))
3		nfv 1619	. . . 4 ⊢ Ⅎxψ
4	3	nfal 1842	. . 3 ⊢ Ⅎx∀yψ
5	4	19.36 1871	. 2 ⊢ (∃x(φ → ∀yψ) ↔ (∀xφ → ∀yψ))
6		19.36v 1896	. . . 4 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ψ))
7	6	albii 1566	. . 3 ⊢ (∀y∃x(φ → ψ) ↔ ∀y(∀xφ → ψ))
8		nfv 1619	. . . . 5 ⊢ Ⅎyφ
9	8	nfal 1842	. . . 4 ⊢ Ⅎy∀xφ
10	9	19.21 1796	. . 3 ⊢ (∀y(∀xφ → ψ) ↔ (∀xφ → ∀yψ))
11	7, 10	bitr2i 241	. 2 ⊢ ((∀xφ → ∀yψ) ↔ ∀y∃x(φ → ψ))
12	2, 5, 11	3bitri 262	1 ⊢ (∃x∀y(φ → ψ) ↔ ∀y∃x(φ → ψ))

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-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)