Theorem r19.9rzv 3645

Description: Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)

Assertion

Ref	Expression
r19.9rzv	⊢ (A ≠ ∅ → (φ ↔ ∃x ∈ A φ))

Distinct variable groups:   x,A   φ,x

Proof of Theorem r19.9rzv

Step	Hyp	Ref	Expression
1		r19.3rzv 3644	. . . 4 ⊢ (A ≠ ∅ → (¬ φ ↔ ∀x ∈ A ¬ φ))
2	1	bicomd 192	. . 3 ⊢ (A ≠ ∅ → (∀x ∈ A ¬ φ ↔ ¬ φ))
3	2	con2bid 319	. 2 ⊢ (A ≠ ∅ → (φ ↔ ¬ ∀x ∈ A ¬ φ))
4		dfrex2 2628	. 2 ⊢ (∃x ∈ A φ ↔ ¬ ∀x ∈ A ¬ φ)
5	3, 4	syl6bbr 254	1 ⊢ (A ≠ ∅ → (φ ↔ ∃x ∈ A φ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ≠ wne 2517  ∀wral 2615  ∃wrex 2616  ∅c0 3551

This theorem was proved from 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 theorem 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-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  r19.45zv  3648  r19.36zv  3651  iunconst  3978  fconstfv  5457