Theorem 19.21-2OLD 2215

Description: Obsolete proof of 19.21-2 2078 as of 6-Oct-2021. (Contributed by NM, 4-Feb-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
19.21-2OLD.1	⊢ Ⅎ𝑥𝜑
19.21-2OLD.2	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
19.21-2OLD	⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))

Proof of Theorem 19.21-2OLD

Step	Hyp	Ref	Expression
1		19.21-2OLD.2	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	19.21OLD 2214	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓))
3	2	albii 1747	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
4		19.21-2OLD.1	. . 3 ⊢ Ⅎ𝑥𝜑
5	4	19.21OLD 2214	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))
6	3, 5	bitri 264	1 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥∀𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  ℲwnfOLD 1709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nfOLD 1721

This theorem is referenced by: (None)