Theorem 19.26-2 1878

Description: Theorem 19.26 1877 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 1877	. . 3 ⊢ (∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑦𝜑 ∧ ∀𝑦𝜓))
2	1	albii 1826	. 2 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ ∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓))
3		19.26 1877	. 2 ⊢ (∀𝑥(∀𝑦𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓))
4	2, 3	bitri 276	1 ⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓))

Syntax hints:   ↔ wb 207   ∧ wa 396  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816

This theorem depends on definitions:  df-bi 208  df-an 397

This theorem is referenced by:  aaan  2341  2mo2  2651  opelopabt  5474  fun11  6559  dford4  43474  undmrnresiss  44048