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Theorem ax11i 1643
Description: Inference that has ax-11 1438 (without  A. y) as its conclusion and doesn't require ax-10 1437, ax-11 1438, or ax-12 1443 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
Hypotheses
Ref Expression
ax11i.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
ax11i.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
ax11i  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )

Proof of Theorem ax11i
StepHypRef Expression
1 ax11i.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
2 ax11i.2 . . 3  |-  ( ps 
->  A. x ps )
31biimprcd 158 . . 3  |-  ( ps 
->  ( x  =  y  ->  ph ) )
42, 3alrimih 1399 . 2  |-  ( ps 
->  A. x ( x  =  y  ->  ph )
)
51, 4syl6bi 161 1  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379
This theorem depends on definitions:  df-bi 115
This theorem is referenced by: (None)
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