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Theorem bj-ax12i 31005
Description: A statement close to the axiom of substitution ax-12 1907. The general statement that ax12i 1788 proves. (Contributed by BJ, 29-Sep-2019.)
Hypotheses
Ref Expression
bj-ax12i.1  |-  ( ph  ->  ( ps  <->  ch )
)
bj-ax12i.2  |-  ( ch 
->  A. x ch )
Assertion
Ref Expression
bj-ax12i  |-  ( ph  ->  ( ps  ->  A. x
( ph  ->  ps )
) )

Proof of Theorem bj-ax12i
StepHypRef Expression
1 bj-ax12i.1 . 2  |-  ( ph  ->  ( ps  <->  ch )
)
2 bj-ax12i.2 . . 3  |-  ( ch 
->  A. x ch )
31biimprcd 228 . . 3  |-  ( ch 
->  ( ph  ->  ps ) )
42, 3alrimih 1689 . 2  |-  ( ch 
->  A. x ( ph  ->  ps ) )
51, 4syl6bi 231 1  |-  ( ph  ->  ( ps  ->  A. x
( ph  ->  ps )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678
This theorem depends on definitions:  df-bi 188
This theorem is referenced by:  bj-ax12wlem  31007
  Copyright terms: Public domain W3C validator