Theorem r19.29r 3217

Description: Restricted quantifier version of 19.29r 1875; variation of r19.29 3216. (Contributed by NM, 31-Aug-1999.) (Proof shortened by Wolf Lammen, 29-Jun-2023.)

Assertion

Ref	Expression
r19.29r	⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Proof of Theorem r19.29r

Step	Hyp	Ref	Expression
1		r19.29 3216	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜓 ∧ ∃𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
2	1	ancoms 462	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
3		pm3.22 463	. . 3 ⊢ ((𝜓 ∧ 𝜑) → (𝜑 ∧ 𝜓))
4	3	reximi 3206	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))
5	2, 4	syl 17	1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 399  ∀wral 3106  ∃wrex 3107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ral 3111  df-rex 3112

This theorem is referenced by:  r19.29imd  3218  2reu5  3697  rlimuni  14899  rlimno1  15002  neindisj2  21728  lmss  21903  fclsbas  22626  isfcf  22639  ucnima  22887  metcnp3  23147  cfilucfil  23166  bndth  23563  ellimc3  24482  lmxrge0  31305  gsumesum  31428  esumcst  31432  esumfsup  31439  voliune  31598  volfiniune  31599  bnj517  32267  nummin  32474  fvineqsneq  34829  cover2  35152  prmunb2  41015