Theorem 19.42-1 2104

Description: One direction of 19.42 2105. (Contributed by Wolf Lammen, 10-Jul-2021.)

Hypothesis

Ref	Expression
19.42.1	⊢ Ⅎ𝑥𝜑

Assertion

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

Proof of Theorem 19.42-1

Step	Hyp	Ref	Expression
1		19.42.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		pm3.2 463	. . 3 ⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
3	1, 2	eximd 2085	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
4	3	imp 445	1 ⊢ ((𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1704  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  bnj596  30816