Theorem 19.41 1879

Description: Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
19.41.1	⊢ Ⅎxψ

Assertion

Ref	Expression
19.41	⊢ (∃x(φ ∧ ψ) ↔ (∃xφ ∧ ψ))

Proof of Theorem 19.41

Step	Hyp	Ref	Expression
1		19.40 1609	. . 3 ⊢ (∃x(φ ∧ ψ) → (∃xφ ∧ ∃xψ))
2		19.41.1	. . . . 5 ⊢ Ⅎxψ
3		id 19	. . . . 5 ⊢ (ψ → ψ)
4	2, 3	exlimi 1803	. . . 4 ⊢ (∃xψ → ψ)
5	4	anim2i 552	. . 3 ⊢ ((∃xφ ∧ ∃xψ) → (∃xφ ∧ ψ))
6	1, 5	syl 15	. 2 ⊢ (∃x(φ ∧ ψ) → (∃xφ ∧ ψ))
7		pm3.21 435	. . . 4 ⊢ (ψ → (φ → (φ ∧ ψ)))
8	2, 7	eximd 1770	. . 3 ⊢ (ψ → (∃xφ → ∃x(φ ∧ ψ)))
9	8	impcom 419	. 2 ⊢ ((∃xφ ∧ ψ) → ∃x(φ ∧ ψ))
10	6, 9	impbii 180	1 ⊢ (∃x(φ ∧ ψ) ↔ (∃xφ ∧ ψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544

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:  19.42  1880  19.41v  1901  eean  1912  r19.41  2764  eliunxp  4822