Theorem 19.40 1920

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 1900	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)
2		exsimpr 1901	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜓)
3	1, 2	jca 521	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  19.40-2  1921  19.40b  1922  19.41v  1983  19.41  2274  exdistrf  2479  uniinOLD  4894  copsexgwOLD  5464  copsexg  5465  dmin  5897  imadif  6632  oprabidw  7458  lfuhgr3  29740  bj-19.41al  37685  bj-nnfan  37783  bj-nnfand  37784  bj-19.42t  37794