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Theorem hbalw 1683
Description: Weak version of hbal 1710. Uses only Tarski's FOL axiom schemes. Unlike hbal 1710, this theorem requires that  x and  y be distinct i.e. are not bundled. (Contributed by NM, 19-Apr-2017.)
Hypotheses
Ref Expression
hbalw.1  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
hbalw.2  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
hbalw  |-  ( A. y ph  ->  A. x A. y ph )
Distinct variable groups:    x, z    x, y    ph, z    ps, x
Allowed substitution hints:    ph( x, y)    ps( y, z)

Proof of Theorem hbalw
StepHypRef Expression
1 hbalw.2 . . 3  |-  ( ph  ->  A. x ph )
21alimi 1546 . 2  |-  ( A. y ph  ->  A. y A. x ph )
3 hbalw.1 . . 3  |-  ( x  =  z  ->  ( ph 
<->  ps ) )
43alcomiw 1678 . 2  |-  ( A. y A. x ph  ->  A. x A. y ph )
52, 4syl 15 1  |-  ( A. y ph  ->  A. x A. y ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527
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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