Theorem 19.42-1OLD 2271

Description: One direction of 19.42 2272. Obsolete as of 9-Oct-2022. (Contributed by Wolf Lammen, 10-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.42.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
19.42-1OLD	⊢ ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Proof of Theorem 19.42-1OLD

Step	Hyp	Ref	Expression
1		19.42.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		pm3.2 462	. . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
3	1, 2	eximd 2251	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
4	3	imp 396	1 ⊢ ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 385  ∃wex 1875  Ⅎwnf 1879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-12 2213

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880

This theorem is referenced by: (None)