Theorem 19.40 1504

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 1490	. 2 ⊢ (∃x(φ ∧ ψ) → ∃xφ)
2		simpr 103	. . 3 ⊢ ((φ ∧ ψ) → ψ)
3	2	eximi 1473	. 2 ⊢ (∃x(φ ∧ ψ) → ∃xψ)
4	1, 3	jca 290	1 ⊢ (∃x(φ ∧ ψ) → (∃xφ ∧ ∃xψ))

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1362

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 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-ial 1409

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  19.40-2  1505  19.41h  1557  19.41  1558  exdistrfor  1663  uniin  3574  copsexg  3955  dmin  4470  imadif  4905  imainlem  4906