Theorem r19.32v 3083

Description: Restricted quantifier version of 19.32v 1869. (Contributed by NM, 25-Nov-2003.)

Assertion

Ref	Expression
r19.32v	|- ( A. x e. A ( ph \/ ps ) <-> ( ph \/ A. x e. A ps ) )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem r19.32v

Step	Hyp	Ref	Expression
1		r19.21v 2960	. 2 |- ( A. x e. A ( -. ph -> ps ) <-> ( -. ph -> A. x e. A ps ) )
2		df-or 385	. . 3 |- ( ( ph \/ ps ) <-> ( -. ph -> ps ) )
3	2	ralbii 2980	. 2 |- ( A. x e. A ( ph \/ ps ) <-> A. x e. A ( -. ph -> ps ) )
4		df-or 385	. 2 |- ( ( ph \/ A. x e. A ps ) <-> ( -. ph -> A. x e. A ps ) )
5	1, 3, 4	3bitr4i 292	1 |- ( A. x e. A ( ph \/ ps ) <-> ( ph \/ A. x e. A ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-ral 2917

This theorem is referenced by:  iinun2  4586  iinuni  4609  axcontlem2  25845  axcontlem7  25850  disjnf  29384  lindslinindsimp2  42252