Theorem r19.9rzv 1768

Description: Restricted quantification of wff not containing quantified variable.

Assertion

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

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

Proof of Theorem r19.9rzv

Step	Hyp	Ref	Expression
1		r19.3rzv 1767	. . . 4 ⊢ (¬ A = ∅ → (¬ φ ↔ ∀x ∈ A ¬ φ))
2	1	bicomd 399	. . 3 ⊢ (¬ A = ∅ → (∀x ∈ A ¬ φ ↔ ¬ φ))
3	2	bicon2d 404	. 2 ⊢ (¬ A = ∅ → (φ ↔ ¬ ∀x ∈ A ¬ φ))
4		dfrex2 1212	. 2 ⊢ (∃x ∈ A φ ↔ ¬ ∀x ∈ A ¬ φ)
5	3, 4	syl6bbr 416	1 ⊢ (¬ A = ∅ → (φ ↔ ∃x ∈ A φ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   = wceq 1091  ∀wral 1201  ∃wrex 1202  ∅c0 1707

This theorem is referenced by:  r19.45zv 1770  r19.36zv 1772  fconstfv 2903

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-rex 1206  df-v 1349  df-dif 1489  df-nul 1708

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