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Theorem ex-natded9.26 25868
Description: Theorem 9.26 of [Clemente] p. 45, translated line by line using an interpretation of natural deduction in Metamath. This proof has some additional complications due to the fact that Metamath's existential elimination rule does not change bound variables, so we need to verify that  x is bound in the conclusion. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
13  E. x A. y ps ( x ,  y )  ( ph  ->  E. x A. y ps ) Given $e.
26 ...|  A. y ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  A. y ps ) ND hypothesis assumption simpr 462. Later statements will have this scope.
37;5,4 ...  ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  ps )  A.E 2,y spsbcd 3313 ( A.E), 5,6. To use it we need a1i 11 and vex 3083. This could be immediately done with 19.21bi 1924, but we want to show the general approach for substitution.
412;8,9,10,11 ...  E. x ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  E. x ps )  E.I 3,a spesbcd 3382 ( E.I), 11. To use it we need sylibr 215, which in turn requires sylib 199 and two uses of sbcid 3316. This could be more immediately done using 19.8a 1912, but we want to show the general approach for substitution.
513;1,2  E. x ps ( x ,  y )  ( ph  ->  E. x ps )  E.E 1,2,4,a exlimdd 2039 ( E.E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1755 and nfe1 1894 (MPE# 1,2)
614  A. y E. x ps ( x ,  y )  ( ph  ->  A. y E. x ps )  A.I 5 alrimiv 1767 ( A.I), 13

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. Below is the final metamath proof (which reorders some steps).

Note that in the original proof,  ps ( x ,  y ) has explicit parameters. In Metamath, these parameters are always implicit, and the parameters upon which a wff variable can depend are recorded in the "allowed substitution hints" below.

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded9.26-2 25869.

(Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by David A. Wheeler, 18-Feb-2017.)

Hypothesis
Ref Expression
ex-natded9.26.1  |-  ( ph  ->  E. x A. y ps )
Assertion
Ref Expression
ex-natded9.26  |-  ( ph  ->  A. y E. x ps )
Distinct variable group:    x, y,
ph
Allowed substitution hints:    ps( x, y)

Proof of Theorem ex-natded9.26
StepHypRef Expression
1 nfv 1755 . . 3  |-  F/ x ph
2 nfe1 1894 . . 3  |-  F/ x E. x ps
3 ex-natded9.26.1 . . 3  |-  ( ph  ->  E. x A. y ps )
4 vex 3083 . . . . . . . 8  |-  y  e. 
_V
54a1i 11 . . . . . . 7  |-  ( (
ph  /\  A. y ps )  ->  y  e. 
_V )
6 simpr 462 . . . . . . 7  |-  ( (
ph  /\  A. y ps )  ->  A. y ps )
75, 6spsbcd 3313 . . . . . 6  |-  ( (
ph  /\  A. y ps )  ->  [. y  /  y ]. ps )
8 sbcid 3316 . . . . . 6  |-  ( [. y  /  y ]. ps  <->  ps )
97, 8sylib 199 . . . . 5  |-  ( (
ph  /\  A. y ps )  ->  ps )
10 sbcid 3316 . . . . 5  |-  ( [. x  /  x ]. ps  <->  ps )
119, 10sylibr 215 . . . 4  |-  ( (
ph  /\  A. y ps )  ->  [. x  /  x ]. ps )
1211spesbcd 3382 . . 3  |-  ( (
ph  /\  A. y ps )  ->  E. x ps )
131, 2, 3, 12exlimdd 2039 . 2  |-  ( ph  ->  E. x ps )
1413alrimiv 1767 1  |-  ( ph  ->  A. y E. x ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   A.wal 1435   E.wex 1657    e. wcel 1872   _Vcvv 3080   [.wsbc 3299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-sbc 3300
This theorem is referenced by: (None)
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