Theorem reximdva0 4355

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 4353	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		reximdva0.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
3	2	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	3	ancld 550	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
5	4	eximdv 1917	. . . 4 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)))
6	5	imp 406	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7	1, 6	sylan2b 594	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
8		df-rex 3071	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
9	7, 8	sylibr 234	1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070  ∅c0 4333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-ne 2941  df-rex 3071  df-dif 3954  df-nul 4334

This theorem is referenced by:  n0snor2el  4833  f1cdmsn  7302  hashgt12el  14461  refun0  23523  cstucnd  24293  supxrnemnf  32772  kerunit  33349  ssdifidllem  33484  ssmxidllem  33501  elpaddn0  39802  thinciso  49117