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Theorem 19.21h 1743
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as " x is not free in  ph." (Contributed by NM, 1-Aug-2017.)
Hypothesis
Ref Expression
19.21h.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
19.21h  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )

Proof of Theorem 19.21h
StepHypRef Expression
1 19.21h.1 . . 3  |-  ( ph  ->  A. x ph )
2 alim 1548 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
31, 2syl5 28 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( ph  ->  A. x ps )
)
4 hba1 1731 . . . 4  |-  ( A. x ps  ->  A. x A. x ps )
51, 4hbim 1737 . . 3  |-  ( (
ph  ->  A. x ps )  ->  A. x ( ph  ->  A. x ps )
)
6 sp 1728 . . . 4  |-  ( A. x ps  ->  ps )
76imim2i 13 . . 3  |-  ( (
ph  ->  A. x ps )  ->  ( ph  ->  ps ) )
85, 7alrimih 1555 . 2  |-  ( (
ph  ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
93, 8impbii 180 1  |-  ( A. x ( ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530
This theorem is referenced by:  hbim1  1744  ax12olem6  1885  ax12olem6NEW7  29436  a12study9  29742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177
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