Theorem 19.27 2219

Description: Theorem 19.27 of [Margaris] p. 90. See 19.27v 1992 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)

Hypothesis

Ref	Expression
19.27.1	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
19.27	⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))

Proof of Theorem 19.27

Step	Hyp	Ref	Expression
1		19.26 1872	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.27.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	19.3 2194	. . 3 ⊢ (∀𝑥𝜓 ↔ 𝜓)
4	3	anbi2i 622	. 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
5	1, 4	bitri 275	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 395  ∀wal 1538  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-nf 1785

This theorem is referenced by:  aaanOLD  2327