Theorem 19.26-2 1898

Description: Theorem 19.26 1897 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 1897	. . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑦𝜑 ∧ ∀𝑦𝜓))
2	1	albii 1846	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓))
3		19.26 1897	. 2 ⊢ (∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓))
4	2, 3	bitri 278	1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∀wal 1565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401

This theorem is referenced by:  aaan  2371  2mo2  2681  opelopabt  5517  fun11  6611  dford4  43647  undmrnresiss  44221