Theorem exan 1895

Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021.) (Proof shortened by Wolf Lammen, 6-Nov-2022.) Expand hypothesis. (Revised by Steven Nguyen, 19-Jun-2023.)

Hypotheses

Ref	Expression
exan.1	⊢ ∃𝑥𝜑
exan.2	⊢ 𝜓

Assertion

Ref	Expression
exan	⊢ ∃𝑥(𝜑 ∧ 𝜓)

Proof of Theorem exan

Step	Hyp	Ref	Expression
1		exan.1	. 2 ⊢ ∃𝑥𝜑
2		exan.2	. . 3 ⊢ 𝜓
3	2	jctr 534	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝜓))
4	1, 3	eximii 1870	1 ⊢ ∃𝑥(𝜑 ∧ 𝜓)

This proof depends on syntax axioms:   ∧ wa 401  ∃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:  ac6s6f  39085  fnchoice  46015