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Theorem hba1w 1693
Description: Weak version of hba1 1731. See comments for ax6w 1703. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
Hypothesis
Ref Expression
hbn1w.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
hba1w  |-  ( A. x ph  ->  A. x A. x ph )
Distinct variable groups:    ph, y    ps, x    x, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem hba1w
StepHypRef Expression
1 hbn1w.1 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21cbvalvw 1688 . . . . . 6  |-  ( A. x ph  <->  A. y ps )
32a1i 10 . . . . 5  |-  ( x  =  y  ->  ( A. x ph  <->  A. y ps ) )
43notbid 285 . . . 4  |-  ( x  =  y  ->  ( -.  A. x ph  <->  -.  A. y ps ) )
54spw 1679 . . 3  |-  ( A. x  -.  A. x ph  ->  -.  A. x ph )
65con2i 112 . 2  |-  ( A. x ph  ->  -.  A. x  -.  A. x ph )
74hbn1w 1692 . 2  |-  ( -. 
A. x  -.  A. x ph  ->  A. x  -.  A. x  -.  A. x ph )
81hbn1w 1692 . . . 4  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
98con1i 121 . . 3  |-  ( -. 
A. x  -.  A. x ph  ->  A. x ph )
109alimi 1549 . 2  |-  ( A. x  -.  A. x  -.  A. x ph  ->  A. x A. x ph )
116, 7, 103syl 18 1  |-  ( A. x ph  ->  A. x A. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176   A.wal 1530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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