Theorem 19.21-2 2209

Description: Version of 19.21 2207 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 2207	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓))
3	2	albii 1819	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
4		19.21-2.1	. . 3 ⊢ Ⅎ𝑥𝜑
5	4	19.21 2207	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))
6	3, 5	bitri 275	1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-12 2177

This theorem depends on definitions:  df-bi 207  df-ex 1780  df-nf 1784

This theorem is referenced by:  cotr2g  15015  dford4  43041