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Theorem bj-ceqsalt1 31527
Description: The FOL content of ceqsalt 3081. Lemma for bj-ceqsalt 31528 and bj-ceqsaltv 31529. (TODO: consider removing if it does not add anything to bj-ceqsalt0 31526.) (Contributed by BJ, 26-Sep-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-ceqsalt1.1  |-  ( th 
->  E. x ch )
Assertion
Ref Expression
bj-ceqsalt1  |-  ( ( F/ x ps  /\  A. x ( ch  ->  (
ph 
<->  ps ) )  /\  th )  ->  ( A. x ( ch  ->  ph )  <->  ps ) )

Proof of Theorem bj-ceqsalt1
StepHypRef Expression
1 bj-ceqsalt1.1 . . . 4  |-  ( th 
->  E. x ch )
213ad2ant3 1037 . . 3  |-  ( ( F/ x ps  /\  A. x ( ch  ->  (
ph 
<->  ps ) )  /\  th )  ->  E. x ch )
3 biimp 198 . . . . . . 7  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
43imim3i 61 . . . . . 6  |-  ( ( ch  ->  ( ph  <->  ps ) )  ->  (
( ch  ->  ph )  ->  ( ch  ->  ps ) ) )
54al2imi 1697 . . . . 5  |-  ( A. x ( ch  ->  (
ph 
<->  ps ) )  -> 
( A. x ( ch  ->  ph )  ->  A. x ( ch  ->  ps ) ) )
653ad2ant2 1036 . . . 4  |-  ( ( F/ x ps  /\  A. x ( ch  ->  (
ph 
<->  ps ) )  /\  th )  ->  ( A. x ( ch  ->  ph )  ->  A. x
( ch  ->  ps ) ) )
7 19.23t 2001 . . . . 5  |-  ( F/ x ps  ->  ( A. x ( ch  ->  ps )  <->  ( E. x ch  ->  ps ) ) )
873ad2ant1 1035 . . . 4  |-  ( ( F/ x ps  /\  A. x ( ch  ->  (
ph 
<->  ps ) )  /\  th )  ->  ( A. x ( ch  ->  ps )  <->  ( E. x ch  ->  ps ) ) )
96, 8sylibd 222 . . 3  |-  ( ( F/ x ps  /\  A. x ( ch  ->  (
ph 
<->  ps ) )  /\  th )  ->  ( A. x ( ch  ->  ph )  ->  ( E. x ch  ->  ps )
) )
102, 9mpid 42 . 2  |-  ( ( F/ x ps  /\  A. x ( ch  ->  (
ph 
<->  ps ) )  /\  th )  ->  ( A. x ( ch  ->  ph )  ->  ps )
)
11 biimpr 203 . . . . . . 7  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
1211imim2i 16 . . . . . 6  |-  ( ( ch  ->  ( ph  <->  ps ) )  ->  ( ch  ->  ( ps  ->  ph ) ) )
1312com23 81 . . . . 5  |-  ( ( ch  ->  ( ph  <->  ps ) )  ->  ( ps  ->  ( ch  ->  ph ) ) )
1413alimi 1694 . . . 4  |-  ( A. x ( ch  ->  (
ph 
<->  ps ) )  ->  A. x ( ps  ->  ( ch  ->  ph ) ) )
15143ad2ant2 1036 . . 3  |-  ( ( F/ x ps  /\  A. x ( ch  ->  (
ph 
<->  ps ) )  /\  th )  ->  A. x
( ps  ->  ( ch  ->  ph ) ) )
16 19.21t 1996 . . . 4  |-  ( F/ x ps  ->  ( A. x ( ps  ->  ( ch  ->  ph ) )  <-> 
( ps  ->  A. x
( ch  ->  ph )
) ) )
17163ad2ant1 1035 . . 3  |-  ( ( F/ x ps  /\  A. x ( ch  ->  (
ph 
<->  ps ) )  /\  th )  ->  ( A. x ( ps  ->  ( ch  ->  ph ) )  <-> 
( ps  ->  A. x
( ch  ->  ph )
) ) )
1815, 17mpbid 215 . 2  |-  ( ( F/ x ps  /\  A. x ( ch  ->  (
ph 
<->  ps ) )  /\  th )  ->  ( ps  ->  A. x ( ch 
->  ph ) ) )
1910, 18impbid 195 1  |-  ( ( F/ x ps  /\  A. x ( ch  ->  (
ph 
<->  ps ) )  /\  th )  ->  ( A. x ( ch  ->  ph )  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ w3a 991   A.wal 1452   E.wex 1673   F/wnf 1677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943
This theorem depends on definitions:  df-bi 190  df-an 377  df-3an 993  df-ex 1674  df-nf 1678
This theorem is referenced by: (None)
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