Theorem reximdva0 4307

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 4305	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
2		reximdva0.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓)
3	2	ex 416	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	3	ancld 558	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝜓)))
5	4	eximdv 1936	. . . 4 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)))
6	5	imp 410	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 𝑥 ∈ 𝐴) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7	1, 6	sylan2b 603	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
8		df-rex 3086	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
9	7, 8	sylibr 236	1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1798   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-ne 2957  df-rex 3086  df-dif 3907  df-nul 4286

This theorem is referenced by:  n0snor2el  4790  f1cdmsn  7262  hashgt12el  14432  refun0  23555  cstucnd  24323  supxrnemnf  32920  kerunit  33472  ssdifidllem  33604  ssmxidllem  33622  constrfiss  34009  elpaddn0  40388  nelsubclem  49652  thinciso  50055