Theorem reximdva0 4318

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 4315	. . 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 1944	. . . 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 3096	. 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 1806   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-ne 2965  df-rex 3096  df-dif 3916  df-nul 4295

This theorem is referenced by:  n0snor2el  4802  f1cdmsn  7281  hashgt12el  14459  ssdifidllem  21453  refun0  23641  cstucnd  24409  supxrnemnf  33054  kerunit  33588  ssmxidllem  33701  constrfiss  34086  elpaddn0  40498  nelsubclem  49764  thinciso  50167