Theorem exintr 1892

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 544	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1832	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780

This theorem is referenced by:  equs4v  1999  equs4  2421  eupickbi  2636  barbarilem  2668  ceqsexOLD  3531  r19.2z  4495  pwpw0  4813  bnj1023  34794  bnj1109  34800  pm10.55  44388