Theorem 19.29r 1597

Description: Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Assertion

Ref	Expression
19.29r	⊢ ((∃xφ ∧ ∀xψ) → ∃x(φ ∧ ψ))

Proof of Theorem 19.29r

Step	Hyp	Ref	Expression
1		19.29 1596	. . 3 ⊢ ((∀xψ ∧ ∃xφ) → ∃x(ψ ∧ φ))
2	1	ancoms 439	. 2 ⊢ ((∃xφ ∧ ∀xψ) → ∃x(ψ ∧ φ))
3		exancom 1586	. 2 ⊢ (∃x(φ ∧ ψ) ↔ ∃x(ψ ∧ φ))
4	2, 3	sylibr 203	1 ⊢ ((∃xφ ∧ ∀xψ) → ∃x(φ ∧ ψ))

Syntax hints:   → wi 4   ∧ 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

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

This theorem is referenced by:  19.29r2  1598  19.29x  1599  exan  1882  equvini  1987  eu2  2229  intab  3957  imadif  5172  ncssfin  6152