Theorem r19.32v 2758

Description: Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers. (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 2702	. 2 |- (A.x e. A (-. ph -> ps) <-> (-. ph -> A.x e. A ps))
2		df-or 359	. . 3 |- ((ph \/ ps) <-> (-. ph -> ps))
3	2	ralbii 2639	. 2 |- (A.x e. A (ph \/ ps) <-> A.x e. A (-. ph -> ps))
4		df-or 359	. 2 |- ((ph \/ A.x e. A ps) <-> (-. ph -> A.x e. A ps))
5	1, 3, 4	3bitr4i 268	1 |- (A.x e. A (ph \/ ps) <-> (ph \/ A.x e. A ps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357  A.wral 2615

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-11 1746

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  iinun2  4033  iinuni  4050