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Theorem ex-natded9.26 21568
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 448. Later statements will have this scope.
37;5,4 ...  ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  ps )  A.E 2,y spsbcd 3110 ( A.E), 5,6. To use it we need a1i 11 and vex 2895. This could be immediately done with 19.21bi 1766, 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 3179 ( E.I), 11. To use it we need sylibr 204, which in turn requires sylib 189 and two uses of sbcid 3113. This could be more immediately done using 19.8a 1754, 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 1901 ( E.E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1626 and nfe1 1739 (MPE# 1,2)
614  A. y E. x ps ( x ,  y )  ( ph  ->  A. y E. x ps )  A.I 5 alrimiv 1638 ( 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 21569.

(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 1626 . . 3  |-  F/ x ph
2 nfe1 1739 . . 3  |-  F/ x E. x ps
3 ex-natded9.26.1 . . 3  |-  ( ph  ->  E. x A. y ps )
4 vex 2895 . . . . . . . 8  |-  y  e. 
_V
54a1i 11 . . . . . . 7  |-  ( (
ph  /\  A. y ps )  ->  y  e. 
_V )
6 simpr 448 . . . . . . 7  |-  ( (
ph  /\  A. y ps )  ->  A. y ps )
75, 6spsbcd 3110 . . . . . 6  |-  ( (
ph  /\  A. y ps )  ->  [. y  /  y ]. ps )
8 sbcid 3113 . . . . . 6  |-  ( [. y  /  y ]. ps  <->  ps )
97, 8sylib 189 . . . . 5  |-  ( (
ph  /\  A. y ps )  ->  ps )
10 sbcid 3113 . . . . 5  |-  ( [. x  /  x ]. ps  <->  ps )
119, 10sylibr 204 . . . 4  |-  ( (
ph  /\  A. y ps )  ->  [. x  /  x ]. ps )
1211spesbcd 3179 . . 3  |-  ( (
ph  /\  A. y ps )  ->  E. x ps )
131, 2, 3, 12exlimdd 1901 . 2  |-  ( ph  ->  E. x ps )
1413alrimiv 1638 1  |-  ( ph  ->  A. y E. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547    e. wcel 1717   _Vcvv 2892   [.wsbc 3097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rex 2648  df-v 2894  df-sbc 3098
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