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Theorem ax12i 1807
Description: Inference that has ax-12 1944 (without  A. y) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 1944 in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
Hypotheses
Ref Expression
ax12i.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
ax12i.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
ax12i  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )

Proof of Theorem ax12i
StepHypRef Expression
1 ax12i.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
2 ax12i.2 . . 3  |-  ( ps 
->  A. x ps )
31biimprcd 233 . . 3  |-  ( ps 
->  ( x  =  y  ->  ph ) )
42, 3alrimih 1704 . 2  |-  ( ps 
->  A. x ( x  =  y  ->  ph )
)
51, 4syl6bi 236 1  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693
This theorem depends on definitions:  df-bi 190
This theorem is referenced by:  ax12wlem  1917
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