Theorem exintr 1896

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 546	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜑 ∧ 𝜓)))
2	1	aleximi 1835	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓)))

Syntax hints:   → wi 4   ∧ wa 397  ∀wal 1540  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783

This theorem is referenced by:  equs4v  2004  equs4  2416  eupickbi  2633  barbarilem  2664  ceqsexOLD  3525  ceqsexvOLD  3527  r19.2z  4495  pwpw0  4817  pwsnOLD  4902  bnj1023  33791  bnj1109  33797  pm10.55  43128