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Theorem sb5ALTVD 27969
Description: The following User's Proof is a Natural Deduction Sequent Calculus transcription of the Fitch-style Natural Deduction proof of Unit 20 Excercise 3.a., which is sb5 2039, found in the "Answers to Starred Exercises" on page 457 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. The same proof may also be interpreted as a Virtual Deduction Hilbert-style axiomatic proof. It was completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. sb5ALT 27571 is sb5ALTVD 27969 without virtual deductions and was automatically derived from sb5ALTVD 27969.
1::  |-  (. [ y  /  x ] ph  ->.  [ y  /  x ] ph ).
2::  |-  [ y  /  x ] x  =  y
3:1,2:  |-  (. [ y  /  x ] ph  ->.  [ y  /  x ] ( x  =  y  /\  ph ) ).
4:3:  |-  (. [ y  /  x ] ph  ->.  E. x ( x  =  y  /\  ph  ) ).
5:4:  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph )  )
6::  |-  (. E. x ( x  =  y  /\  ph )  ->.  E. x ( x  =  y  /\  ph ) ).
7::  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  ( x  =  y  /\  ph ) ).
8:7:  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  ph ).
9:7:  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  x  =  y ).
10:8,9:  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph  )  ->.  [ y  /  x ] ph ).
101::  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
11:101,10:  |-  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph  )
12:5,11:  |-  ( ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph  ) )  /\  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
qed:12:  |-  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph )  )
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sb5ALTVD  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sb5ALTVD
StepHypRef Expression
1 idn1 27625 . . . . . 6  |-  (. [
y  /  x ] ph  ->.  [ y  /  x ] ph ).
2 equsb1 1979 . . . . . 6  |-  [ y  /  x ] x  =  y
3 sban 2008 . . . . . . 7  |-  ( [ y  /  x ]
( x  =  y  /\  ph )  <->  ( [
y  /  x ]
x  =  y  /\  [ y  /  x ] ph ) )
43simplbi2com 1364 . . . . . 6  |-  ( [ y  /  x ] ph  ->  ( [ y  /  x ] x  =  y  ->  [ y  /  x ] ( x  =  y  /\  ph ) ) )
51, 2, 4e10 27747 . . . . 5  |-  (. [
y  /  x ] ph  ->.  [ y  /  x ] ( x  =  y  /\  ph ) ).
6 spsbe 2014 . . . . 5  |-  ( [ y  /  x ]
( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  ph )
)
75, 6e1_ 27679 . . . 4  |-  (. [
y  /  x ] ph  ->.  E. x ( x  =  y  /\  ph ) ).
87in1 27622 . . 3  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
9 hbs1 2044 . . . 4  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
10 idn2 27665 . . . . . 6  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph )  ->.  ( x  =  y  /\  ph ) ).
11 simpr 447 . . . . . 6  |-  ( ( x  =  y  /\  ph )  ->  ph )
1210, 11e2 27683 . . . . 5  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph )  ->.  ph ).
13 simpl 443 . . . . . 6  |-  ( ( x  =  y  /\  ph )  ->  x  =  y )
1410, 13e2 27683 . . . . 5  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph )  ->.  x  =  y ).
15 sbequ1 1861 . . . . . 6  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
1615com12 27 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  [ y  /  x ] ph ) )
1712, 14, 16e22 27723 . . . 4  |-  (. E. x ( x  =  y  /\  ph ) ,. ( x  =  y  /\  ph )  ->.  [ y  /  x ] ph ).
189, 17exinst 27676 . . 3  |-  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph )
198, 18pm3.2i 441 . 2  |-  ( ( [ y  /  x ] ph  ->  E. x
( x  =  y  /\  ph ) )  /\  ( E. x
( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
20 bi3 179 . . 3  |-  ( ( [ y  /  x ] ph  ->  E. x
( x  =  y  /\  ph ) )  ->  ( ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph )  ->  ( [ y  /  x ] ph  <->  E. x ( x  =  y  /\  ph ) ) ) )
2120imp 418 . 2  |-  ( ( ( [ y  /  x ] ph  ->  E. x
( x  =  y  /\  ph ) )  /\  ( E. x
( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )  -> 
( [ y  /  x ] ph  <->  E. x
( x  =  y  /\  ph ) ) )
2219, 21e0_ 27827 1  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623   [wsb 1630
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 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-vd1 27621  df-vd2 27630
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