Theorem 19.40 1684

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 1670	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)
2		simpr 110	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
3	2	eximi 1653	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜓)
4	1, 3	jca 306	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  ∃wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117

This theorem is used by:  19.40-2  1685  19.41h  1737  19.41  1738  exdistrfor  1853  uniin  3955  copsexg  4384  dmin  4989  imadif  5461  imainlem  5462