Theorem 19.21-2 2245

Description: Version of 19.21 2243 with two 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 2243	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓))
3	2	albii 1851	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
4		19.21-2.1	. . 3 ⊢ Ⅎ𝑥𝜑
5	4	19.21 2243	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))
6	3, 5	bitri 278	1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816

This theorem is used by:  cotr2g  15138  dford4  44173