Theorem exan 1882

Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
exan.1	⊢ (∃xφ ∧ ψ)

Assertion

Ref	Expression
exan	⊢ ∃x(φ ∧ ψ)

Proof of Theorem exan

Step	Hyp	Ref	Expression
1		nfe1 1732	. . . 4 ⊢ Ⅎx∃xφ
2	1	19.28 1870	. . 3 ⊢ (∀x(∃xφ ∧ ψ) ↔ (∃xφ ∧ ∀xψ))
3		exan.1	. . 3 ⊢ (∃xφ ∧ ψ)
4	2, 3	mpgbi 1549	. 2 ⊢ (∃xφ ∧ ∀xψ)
5		19.29r 1597	. 2 ⊢ ((∃xφ ∧ ∀xψ) → ∃x(φ ∧ ψ))
6	4, 5	ax-mp 5	1 ⊢ ∃x(φ ∧ ψ)

Syntax hints:   ∧ wa 358  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)