Theorem 19.26-2 1875

Description: Theorem 19.26 1874 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 1874	. . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑦𝜑 ∧ ∀𝑦𝜓))
2	1	albii 1823	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓))
3		19.26 1874	. 2 ⊢ (∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓))
4	2, 3	bitri 274	1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-an 396

This theorem is referenced by:  2mo2  2649  opelopabt  5438  fun11  6492  dford4  40767  undmrnresiss  41101