Theorem 19.21-2 1864

Description: Theorem 19.21 of [Margaris] p. 90 but with 2 quantifiers. (Contributed by NM, 4-Feb-2005.)

Hypotheses

Ref	Expression
19.21-2.1	⊢ Ⅎxφ
19.21-2.2	⊢ Ⅎyφ

Assertion

Ref	Expression
19.21-2	⊢ (∀x∀y(φ → ψ) ↔ (φ → ∀x∀yψ))

Proof of Theorem 19.21-2

Step	Hyp	Ref	Expression
1		19.21-2.2	. . . 4 ⊢ Ⅎyφ
2	1	19.21 1796	. . 3 ⊢ (∀y(φ → ψ) ↔ (φ → ∀yψ))
3	2	albii 1566	. 2 ⊢ (∀x∀y(φ → ψ) ↔ ∀x(φ → ∀yψ))
4		19.21-2.1	. . 3 ⊢ Ⅎxφ
5	4	19.21 1796	. 2 ⊢ (∀x(φ → ∀yψ) ↔ (φ → ∀x∀yψ))
6	3, 5	bitri 240	1 ⊢ (∀x∀y(φ → ψ) ↔ (φ → ∀x∀yψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544

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-11 1746

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

This theorem is referenced by:  2eu6  2289