Theorem r19.32 41167

Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3083. (Contributed by Alexander van der Vekens, 29-Jun-2017.)

Hypothesis

Ref	Expression
r19.32.1	|- F/ x ph

Assertion

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

Proof of Theorem r19.32

Step	Hyp	Ref	Expression
1		r19.32.1	. . . 4 |- F/ x ph
2	1	nfn 1784	. . 3 |- F/ x -. ph
3	2	r19.21 2956	. 2 |- ( A. x e. A ( -. ph -> ps ) <-> ( -. ph -> A. x e. A ps ) )
4		df-or 385	. . 3 |- ( ( ph \/ ps ) <-> ( -. ph -> ps ) )
5	4	ralbii 2980	. 2 |- ( A. x e. A ( ph \/ ps ) <-> A. x e. A ( -. ph -> ps ) )
6		df-or 385	. 2 |- ( ( ph \/ A. x e. A ps ) <-> ( -. ph -> A. x e. A ps ) )
7	3, 5, 6	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   F/wnf 1708   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  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710  df-ral 2917

This theorem is referenced by:  2reu3  41188