Theorem 19.40 1919

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 1901	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)
2		exsimpr 1902	. 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 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  19.40-2  1920  19.40b  1921  19.41v  1982  19.41  2271  exdistrf  2476  uniinOLD  4892  copsexgwOLD  5467  copsexg  5468  dmin  5895  imadif  6618  oprabidw  7445  lfuhgr3  29610  bj-19.41al  37392  bj-nnfan  37490  bj-nnfand  37491  bj-19.42t  37501