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Theorem ax7wK 27819
Description: Weak version of ax-7 1535 from which we can prove any ax-7 1535 instance not involving wff variables or bundling. 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. Unlike ax-7 1535, this theorem requires that  x and  y be distinct i.e. are not bundled. See the description in the comment of equidK 27792. (Contributed by NM, 10-Apr-2017.)
Hypothesis
Ref Expression
ax7wK.1  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
ax7wK  |-  ( A. x A. y ph  ->  A. y A. x ph )
Distinct variable groups:    y, z    x, y    ph, z    ps, y
Allowed substitution hints:    ph( x, y)    ps( x, z)

Proof of Theorem ax7wK
StepHypRef Expression
1 ax7wK.1 . . . . 5  |-  ( y  =  z  ->  ( ph 
<->  ps ) )
21biimpd 200 . . . 4  |-  ( y  =  z  ->  ( ph  ->  ps ) )
32cbvalivK 27811 . . 3  |-  ( A. y ph  ->  A. z ps )
43alimiK 27798 . 2  |-  ( A. x A. y ph  ->  A. x A. z ps )
5 ax-17 1628 . 2  |-  ( A. x A. z ps  ->  A. y A. x A. z ps )
6 equcomiK 27793 . . . . . 6  |-  ( z  =  y  ->  y  =  z )
71biimprd 216 . . . . . 6  |-  ( y  =  z  ->  ( ps  ->  ph ) )
86, 7syl 17 . . . . 5  |-  ( z  =  y  ->  ( ps  ->  ph ) )
98a4imvK 27805 . . . 4  |-  ( A. z ps  ->  ph )
109alimiK 27798 . . 3  |-  ( A. x A. z ps  ->  A. x ph )
1110alimiK 27798 . 2  |-  ( A. y A. x A. z ps  ->  A. y A. x ph )
124, 5, 113syl 20 1  |-  ( A. x A. y ph  ->  A. y A. x ph )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   A.wal 1532    = wceq 1619
This theorem is referenced by:  hbalwK  27820
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-8 1623  ax-17 1628  ax-9v 1632
This theorem depends on definitions:  df-bi 179
  Copyright terms: Public domain W3C validator