Theorem 19.26-2 1594

Description: Theorem 19.26 of [Margaris] p. 90 with two quantifiers. (Contributed by NM, 3-Feb-2005.)

Assertion

Ref	Expression
19.26-2	⊢ (∀x∀y(φ ∧ ψ) ↔ (∀x∀yφ ∧ ∀x∀yψ))

Proof of Theorem 19.26-2

Step	Hyp	Ref	Expression
1		19.26 1593	. . 3 ⊢ (∀y(φ ∧ ψ) ↔ (∀yφ ∧ ∀yψ))
2	1	albii 1566	. 2 ⊢ (∀x∀y(φ ∧ ψ) ↔ ∀x(∀yφ ∧ ∀yψ))
3		19.26 1593	. 2 ⊢ (∀x(∀yφ ∧ ∀yψ) ↔ (∀x∀yφ ∧ ∀x∀yψ))
4	2, 3	bitri 240	1 ⊢ (∀x∀y(φ ∧ ψ) ↔ (∀x∀yφ ∧ ∀x∀yψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  opelopabt  4700  fun11  5160