Theorem 19.28 2228

Description: Theorem 19.28 of [Margaris] p. 90. See 19.28v 1990 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 1870	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.28.1	. . 3 ⊢ Ⅎ𝑥𝜑
3	2	19.3 2202	. 2 ⊢ (∀𝑥𝜑 ↔ 𝜑)
4	1, 3	bianbi 627	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∀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-an 396  df-ex 1780  df-nf 1784

This theorem is referenced by:  aaanOLD  2334  wl-ax11-lem7  37592