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Theorem ax11w 1695
Description: Weak version of ax-11 1715 from which we can prove any ax-11 1715 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. An instance of the first hypothesis will normally require that  x and  y be distinct (unless  x does not occur in  ph). (Contributed by NM, 10-Apr-2017.)
Hypotheses
Ref Expression
ax11w.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
ax11w.2  |-  ( y  =  z  ->  ( ph 
<->  ch ) )
Assertion
Ref Expression
ax11w  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Distinct variable groups:    y, z    ps, x    ph, z    ch, y
Allowed substitution hints:    ph( x, y)    ps( y, z)    ch( x, z)

Proof of Theorem ax11w
StepHypRef Expression
1 ax11w.2 . . 3  |-  ( y  =  z  ->  ( ph 
<->  ch ) )
21spw 1660 . 2  |-  ( A. y ph  ->  ph )
3 ax11w.1 . . 3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
43ax11wlem 1694 . 2  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
52, 4syl5 28 1  |-  ( x  =  y  ->  ( A. y ph  ->  A. x
( x  =  y  ->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527
This theorem is referenced by:  ax11wdemo  1697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529
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