Theorem r19.3rzv 1767

Description: Restricted quantification of wff not containing quantified variable.

Assertion

Ref	Expression
r19.3rzv	⊢ (¬ A = ∅ → (φ ↔ ∀x ∈ A φ))

Distinct variable group(s):   x,A   φ,x

Proof of Theorem r19.3rzv

Step	Hyp	Ref	Expression
1		n0 1714	. . 3 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
2		biimt 549	. . 3 ⊢ (∃x x ∈ A → (φ ↔ (∃x x ∈ A → φ)))
3	1, 2	sylbi 174	. 2 ⊢ (¬ A = ∅ → (φ ↔ (∃x x ∈ A → φ)))
4		df-ral 1205	. . 3 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
5		19.23v 950	. . 3 ⊢ (∀x(x ∈ A → φ) ↔ (∃x x ∈ A → φ))
6	4, 5	bitr 151	. 2 ⊢ (∀x ∈ A φ ↔ (∃x x ∈ A → φ))
7	3, 6	syl6bbr 416	1 ⊢ (¬ A = ∅ → (φ ↔ ∀x ∈ A φ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∅c0 1707

This theorem is referenced by:  r19.9rzv 1768  r19.28zv 1769  r19.27zv 1771  raaan 1775  fint 2769

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-nul 1708

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