Theorem 19.40 1887

Description: Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)

Assertion

Ref	Expression
19.40	⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

Proof of Theorem 19.40

Step	Hyp	Ref	Expression
1		exsimpl 1869	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)
2		exsimpr 1870	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜓)
3	1, 2	jca 511	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1780

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by:  19.40-2  1888  19.40b  1889  19.41v  1950  19.41  2242  exdistrf  2451  uniin  4887  copsexgw  5438  copsexg  5439  dmin  5860  imadif  6576  oprabidw  7389  lfuhgr3  35314  bj-19.41al  36860  bj-nnfan  36949  bj-nnfand  36950  bj-19.42t  36974