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Theorem ax12wlem 1880
Description: Lemma for weak version of ax-12 1907. Uses only Tarski's FOL axiom schemes. In some cases, this lemma may lead to shorter proofs than ax12w 1881. (Contributed by NM, 10-Apr-2017.)
Hypothesis
Ref Expression
ax12wlemw.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ax12wlem  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Distinct variable group:    ps, x
Allowed substitution hints:    ph( x, y)    ps( y)

Proof of Theorem ax12wlem
StepHypRef Expression
1 ax12wlemw.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
2 ax-5 1751 . 2  |-  ( ps 
->  A. x ps )
31, 2ax12i 1788 1  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
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  ax-5 1751
This theorem depends on definitions:  df-bi 188
This theorem is referenced by:  ax12w  1881
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