Theorem reximdva0 3562

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 3560	. . 3 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A)
2		reximdva0.1	. . . . . . 7 ⊢ ((φ ∧ x ∈ A) → ψ)
3	2	ex 423	. . . . . 6 ⊢ (φ → (x ∈ A → ψ))
4	3	ancld 536	. . . . 5 ⊢ (φ → (x ∈ A → (x ∈ A ∧ ψ)))
5	4	eximdv 1622	. . . 4 ⊢ (φ → (∃x x ∈ A → ∃x(x ∈ A ∧ ψ)))
6	5	imp 418	. . 3 ⊢ ((φ ∧ ∃x x ∈ A) → ∃x(x ∈ A ∧ ψ))
7	1, 6	sylan2b 461	. 2 ⊢ ((φ ∧ A ≠ ∅) → ∃x(x ∈ A ∧ ψ))
8		df-rex 2621	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ))
9	7, 8	sylibr 203	1 ⊢ ((φ ∧ A ≠ ∅) → ∃x ∈ A ψ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358  ∃wex 1541   ∈ wcel 1710   ≠ wne 2517  ∃wrex 2616  ∅c0 3551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is used by: (None)