Theorem exan 1775

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, 8-Oct-2021.)

Hypothesis

Ref	Expression
exan.1	⊢ (∃𝑥𝜑 ∧ 𝜓)

Assertion

Ref	Expression
exan	⊢ ∃𝑥(𝜑 ∧ 𝜓)

Proof of Theorem exan

Step	Hyp	Ref	Expression
1		exan.1	. . 3 ⊢ (∃𝑥𝜑 ∧ 𝜓)
2	1	simpli 473	. 2 ⊢ ∃𝑥𝜑
3	1	simpri 477	. . . 4 ⊢ 𝜓
4		pm3.21 463	. . . 4 ⊢ (𝜓 → (𝜑 → (𝜑 ∧ 𝜓)))
5	3, 4	ax-mp 5	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝜓))
6	5	eximi 1752	. 2 ⊢ (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))
7	2, 6	ax-mp 5	1 ⊢ ∃𝑥(𝜑 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  bm1.3ii  4712  ac6s6f  33151  fnchoice  38211