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Theorem bnj982 33980
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj982.1  |-  ( ph  ->  A. x ph )
bnj982.2  |-  ( ps 
->  A. x ps )
bnj982.3  |-  ( ch 
->  A. x ch )
bnj982.4  |-  ( th 
->  A. x th )
Assertion
Ref Expression
bnj982  |-  ( (
ph  /\  ps  /\  ch  /\ 
th )  ->  A. x
( ph  /\  ps  /\  ch  /\  th ) )

Proof of Theorem bnj982
StepHypRef Expression
1 df-bnj17 33882 . 2  |-  ( (
ph  /\  ps  /\  ch  /\ 
th )  <->  ( ( ph  /\  ps  /\  ch )  /\  th ) )
2 bnj982.1 . . . 4  |-  ( ph  ->  A. x ph )
3 bnj982.2 . . . 4  |-  ( ps 
->  A. x ps )
4 bnj982.3 . . . 4  |-  ( ch 
->  A. x ch )
52, 3, 4hb3an 1933 . . 3  |-  ( (
ph  /\  ps  /\  ch )  ->  A. x ( ph  /\ 
ps  /\  ch )
)
6 bnj982.4 . . 3  |-  ( th 
->  A. x th )
75, 6hban 1932 . 2  |-  ( ( ( ph  /\  ps  /\ 
ch )  /\  th )  ->  A. x ( (
ph  /\  ps  /\  ch )  /\  th ) )
81, 7hbxfrbi 1644 1  |-  ( (
ph  /\  ps  /\  ch  /\ 
th )  ->  A. x
( ph  /\  ps  /\  ch  /\  th ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973   A.wal 1393    /\ w-bnj17 33881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-12 1855
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-ex 1614  df-nf 1618  df-bnj17 33882
This theorem is referenced by:  bnj1096  33984  bnj1311  34223  bnj1445  34243
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