Theorem reximdva0 4311

Description: Restricted existence deduced from nonempty class. (Contributed by NM, 1-Feb-2012.)

Hypothesis

Ref	Expression
reximdva0.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)

Assertion

Ref	Expression
reximdva0	⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 𝜓)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem reximdva0

Step	Hyp	Ref	Expression
1		n0 4308	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		reximdva0.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
3	2	ex 417	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	3	ancld 559	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
5	4	eximdv 1940	. . . 4 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)))
6	5	imp 411	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7	1, 6	sylan2b 605	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
8		df-rex 3090	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
9	7, 8	sylibr 237	1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960  ∃wrex 3089  ∅c0 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-ne 2961  df-rex 3090  df-dif 3910  df-nul 4289

This theorem is referenced by:  n0snor2el  4794  f1cdmsn  7270  hashgt12el  14449  ssdifidllem  21444  refun0  23633  cstucnd  24401  supxrnemnf  33025  kerunit  33560  ssmxidllem  33673  constrfiss  34058  elpaddn0  40436  nelsubclem  49696  thinciso  50099