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Theorem 19.12b 24229
Description: 19.12vv 1851 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
19.12b.1  |-  F/ y
ph
19.12b.2  |-  F/ x ps
Assertion
Ref Expression
19.12b  |-  ( E. x A. y (
ph  ->  ps )  <->  A. y E. x ( ph  ->  ps ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem 19.12b
StepHypRef Expression
1 19.12b.1 . . . 4  |-  F/ y
ph
2119.21 1803 . . 3  |-  ( A. y ( ph  ->  ps )  <->  ( ph  ->  A. y ps ) )
32exbii 1572 . 2  |-  ( E. x A. y (
ph  ->  ps )  <->  E. x
( ph  ->  A. y ps ) )
4 19.12b.2 . . . 4  |-  F/ x ps
54nfal 1778 . . 3  |-  F/ x A. y ps
6519.36 1819 . 2  |-  ( E. x ( ph  ->  A. y ps )  <->  ( A. x ph  ->  A. y ps ) )
7419.36 1819 . . . 4  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  ps ) )
87albii 1556 . . 3  |-  ( A. y E. x ( ph  ->  ps )  <->  A. y
( A. x ph  ->  ps ) )
91nfal 1778 . . . 4  |-  F/ y A. x ph
10919.21 1803 . . 3  |-  ( A. y ( A. x ph  ->  ps )  <->  ( A. x ph  ->  A. y ps ) )
118, 10bitr2i 241 . 2  |-  ( ( A. x ph  ->  A. y ps )  <->  A. y E. x ( ph  ->  ps ) )
123, 6, 113bitri 262 1  |-  ( E. x A. y (
ph  ->  ps )  <->  A. y E. x ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   E.wex 1531   F/wnf 1534
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-7 1720  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535
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