Theorem 19.27 2095

Description: Theorem 19.27 of [Margaris] p. 90. See 19.27v 1908 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 1798	. 2 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
2		19.27.1	. . . 4 ⊢ Ⅎ𝑥𝜓
3	2	19.3 2069	. . 3 ⊢ (∀𝑥𝜓 ↔ 𝜓)
4	3	anbi2i 730	. 2 ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))
5	1, 4	bitri 264	1 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ 𝜓))

Syntax hints:   ↔ wb 196   ∧ wa 384  ∀wal 1481  Ⅎwnf 1708

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-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  aaan  2170