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Theorem ax11wlemK 27823
Description: Lemma for weak version of ax-11 1624. Uses only Tarski's predicate calculus axiom schemes i.e. does not use ax-6 1534, ax-7 1535, ax-11 1624, or ax-12 1633, and uses ax-9v 1632 instead of ax-9 1684. In some cases, this lemma may lead to shorter proofs than ax11wK 27824. (Contributed by NM, 10-Apr-2017.)
Hypothesis
Ref Expression
ax11wlemK.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ax11wlemK  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Distinct variable group:    ps, x
Allowed substitution hints:    ph( x, y)    ps( y)

Proof of Theorem ax11wlemK
StepHypRef Expression
1 ax11wlemK.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21biimpa 472 . . 3  |-  ( ( x  =  y  /\  ph )  ->  ps )
3 ax-17 1628 . . . 4  |-  ( ps 
->  A. x ps )
41biimprcd 218 . . . . 5  |-  ( ps 
->  ( x  =  y  ->  ph ) )
54alimiK 27798 . . . 4  |-  ( A. x ps  ->  A. x
( x  =  y  ->  ph ) )
63, 5syl 17 . . 3  |-  ( ps 
->  A. x ( x  =  y  ->  ph )
)
72, 6syl 17 . 2  |-  ( ( x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) )
87ex 425 1  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532
This theorem is referenced by:  ax11wK  27824  ax12o10lem3K  27836
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-gen 1536  ax-17 1628
This theorem depends on definitions:  df-bi 179  df-an 362
  Copyright terms: Public domain W3C validator