Theorem exintr 1925

Description: Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.) (Proof shortened by BJ, 16-Sep-2022.)

Assertion

Ref	Expression
exintr	⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))

Proof of Theorem exintr

Step	Hyp	Ref	Expression
1		ancl 554	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1865	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  equs4v  2033  equs4  2446  eupickbi  2662  barbarilem  2693  r19.2z  4455  pwpw0  4774  bnj1023  35394  bnj1109  35400  pm10.55  45312