Theorem r19.3rz 3642

Description: Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)

Hypothesis

Ref	Expression
r19.3rz.1	⊢ Ⅎxφ

Assertion

Ref	Expression
r19.3rz	⊢ (A ≠ ∅ → (φ ↔ ∀x ∈ A φ))

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem r19.3rz

Step	Hyp	Ref	Expression
1		n0 3560	. . 3 ⊢ (A ≠ ∅ ↔ ∃x x ∈ A)
2		biimt 325	. . 3 ⊢ (∃x x ∈ A → (φ ↔ (∃x x ∈ A → φ)))
3	1, 2	sylbi 187	. 2 ⊢ (A ≠ ∅ → (φ ↔ (∃x x ∈ A → φ)))
4		df-ral 2620	. . 3 ⊢ (∀x ∈ A φ ↔ ∀x(x ∈ A → φ))
5		r19.3rz.1	. . . 4 ⊢ Ⅎxφ
6	5	19.23 1801	. . 3 ⊢ (∀x(x ∈ A → φ) ↔ (∃x x ∈ A → φ))
7	4, 6	bitri 240	. 2 ⊢ (∀x ∈ A φ ↔ (∃x x ∈ A → φ))
8	3, 7	syl6bbr 254	1 ⊢ (A ≠ ∅ → (φ ↔ ∀x ∈ A φ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   ∈ wcel 1710   ≠ wne 2517  ∀wral 2615  ∅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-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is used by:  r19.28z  3643  r19.27z  3649