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Theorem ex-natded9.26 20808
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 447. Later statements will have this scope.
37;5,4 ...  ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  ps )  A.E 2,y spsbcd 3006 ( A.E), 5,6. To use it we need a1i 10 and vex 2793. This could be immediately done with 19.21bi 1796, 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 3075 ( E.I), 11. To use it we need sylibr 203, which in turn requires sylib 188 and two uses of sbcid 3009. This could be more immediately done using 19.8a 1720, 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 1832 ( E.E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1607 and nfe1 1708 (MPE# 1,2)
614  A. y E. x ps ( x ,  y )  ( ph  ->  A. y E. x ps )  A.I 5 alrimiv 1619 ( 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 20809.

(Proof modification 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 1607 . . 3  |-  F/ x ph
2 nfe1 1708 . . 3  |-  F/ x E. x ps
3 ex-natded9.26.1 . . 3  |-  ( ph  ->  E. x A. y ps )
4 vex 2793 . . . . . . . 8  |-  y  e. 
_V
54a1i 10 . . . . . . 7  |-  ( (
ph  /\  A. y ps )  ->  y  e. 
_V )
6 simpr 447 . . . . . . 7  |-  ( (
ph  /\  A. y ps )  ->  A. y ps )
75, 6spsbcd 3006 . . . . . 6  |-  ( (
ph  /\  A. y ps )  ->  [. y  /  y ]. ps )
8 sbcid 3009 . . . . . 6  |-  ( [. y  /  y ]. ps  <->  ps )
97, 8sylib 188 . . . . 5  |-  ( (
ph  /\  A. y ps )  ->  ps )
10 sbcid 3009 . . . . 5  |-  ( [. x  /  x ]. ps  <->  ps )
119, 10sylibr 203 . . . 4  |-  ( (
ph  /\  A. y ps )  ->  [. x  /  x ]. ps )
1211spesbcd 3075 . . 3  |-  ( (
ph  /\  A. y ps )  ->  E. x ps )
131, 2, 3, 12exlimdd 1832 . 2  |-  ( ph  ->  E. x ps )
1413alrimiv 1619 1  |-  ( ph  ->  A. y E. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1529   E.wex 1530    e. wcel 1686   _Vcvv 2790   [.wsbc 2993
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ral 2550  df-rex 2551  df-v 2792  df-sbc 2994
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