Theorem 19.40 1893

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 1875	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)
2		exsimpr 1876	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜓)
3	1, 2	jca 515	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1787

This theorem is referenced by:  19.40-2  1894  19.40b  1895  19.41v  1957  19.41  2237  exdistrf  2448  uniin  4832  copsexgw  5357  copsexg  5358  dmin  5764  imadif  6434  oprabidw  7214  lfuhgr3  32665  bj-19.41al  34496  bj-nnfan  34586  bj-nnfand  34587  bj-19.42t  34611