Theorem r19.21t 1707

Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version).

Assertion

Ref	Expression
r19.21t	|- (A.x(ph -> A.xph) -> (A.x e. A (ph -> ps) <-> (ph -> A.x e. A ps)))

Proof of Theorem r19.21t

Step	Hyp	Ref	Expression
1		19.21t 1111	. . 3 |- (A.x(ph -> A.xph) -> (A.x(ph -> (x e. A -> ps)) <-> (ph -> A.x(x e. A -> ps))))
2		bi2.04 160	. . . 4 |- ((x e. A -> (ph -> ps)) <-> (ph -> (x e. A -> ps)))
3	2	albii 996	. . 3 |- (A.x(x e. A -> (ph -> ps)) <-> A.x(ph -> (x e. A -> ps)))
4	1, 3	syl5bb 530	. 2 |- (A.x(ph -> A.xph) -> (A.x(x e. A -> (ph -> ps)) <-> (ph -> A.x(x e. A -> ps))))
5		df-ral 1641	. 2 |- (A.x e. A (ph -> ps) <-> A.x(x e. A -> (ph -> ps)))
6		df-ral 1641	. . 3 |- (A.x e. A ps <-> A.x(x e. A -> ps))
7	6	imbi2i 185	. 2 |- ((ph -> A.x e. A ps) <-> (ph -> A.x(x e. A -> ps)))
8	4, 5, 7	3bitr4g 553	1 |- (A.x(ph -> A.xph) -> (A.x e. A (ph -> ps) <-> (ph -> A.x e. A ps)))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   e. wcel 955  A.wral 1637

This theorem is referenced by:  r19.21v 1708  sbcralt 1980  sbcralgf 1982

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 960  ax-4 970  ax-5o 972  ax-6o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1641

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