Theorem 19.28 2242

Description: Theorem 19.28 of [Margaris] p. 90. See 19.28v 2004 for a version requiring fewer axioms. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 7-May-2025.)

Hypothesis

Ref	Expression
19.28.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
19.28	⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Proof of Theorem 19.28

Step	Hyp	Ref	Expression
1		19.26 1878	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.28.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	19.3 2216	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜑)
4	1, 3	bianbi 634	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   ↔ wb 208   ∧ wa 397  ∀wal 1546  Ⅎwnf 1791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-12 2191

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-nf 1792

This theorem is referenced by: (None)