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Theorem bnj1101 29217
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1101.1  |-  E. x
( ph  ->  ps )
bnj1101.2  |-  ( ch 
->  ph )
Assertion
Ref Expression
bnj1101  |-  E. x
( ch  ->  ps )

Proof of Theorem bnj1101
StepHypRef Expression
1 bnj1101.1 . . 3  |-  E. x
( ph  ->  ps )
2 pm3.42 545 . . 3  |-  ( (
ph  ->  ps )  -> 
( ( ch  /\  ph )  ->  ps )
)
31, 2bnj101 29150 . 2  |-  E. x
( ( ch  /\  ph )  ->  ps )
4 bnj1101.2 . . . . 5  |-  ( ch 
->  ph )
54pm4.71i 615 . . . 4  |-  ( ch  <->  ( ch  /\  ph )
)
65imbi1i 317 . . 3  |-  ( ( ch  ->  ps )  <->  ( ( ch  /\  ph )  ->  ps ) )
76exbii 1593 . 2  |-  ( E. x ( ch  ->  ps )  <->  E. x ( ( ch  /\  ph )  ->  ps ) )
83, 7mpbir 202 1  |-  E. x
( ch  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551
This theorem is referenced by:  bnj1110  29413  bnj1128  29421  bnj1145  29424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552
  Copyright terms: Public domain W3C validator