Theorem 19.40 1519

Description: Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem 19.40

Step	Hyp	Ref	Expression
1		exsimpl 1505	. 2 ⊢ (∃x(φ ∧ ψ) → ∃xφ)
2		simpr 103	. . 3 ⊢ ((φ ∧ ψ) → ψ)
3	2	eximi 1488	. 2 ⊢ (∃x(φ ∧ ψ) → ∃xψ)
4	1, 3	jca 290	1 ⊢ (∃x(φ ∧ ψ) → (∃xφ ∧ ∃xψ))

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  19.40-2  1520  19.41h  1572  19.41  1573  exdistrfor  1678  uniin  3591  copsexg  3972  dmin  4486  imadif  4922  imainlem  4923