MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ex-natded9.26 Unicode version

Theorem ex-natded9.26 20822
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 3017 ( A.E), 5,6. To use it we need a1i 10 and vex 2804. This could be immediately done with 19.21bi 1806, 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 3086 ( E.I), 11. To use it we need sylibr 203, which in turn requires sylib 188 and two uses of sbcid 3020. This could be more immediately done using 19.8a 1730, 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 1842 ( E.E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1609 and nfe1 1718 (MPE# 1,2)
614  A. y E. x ps ( x ,  y )  ( ph  ->  A. y E. x ps )  A.I 5 alrimiv 1621 ( 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 20823.

(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 1609 . . 3  |-  F/ x ph
2 nfe1 1718 . . 3  |-  F/ x E. x ps
3 ex-natded9.26.1 . . 3  |-  ( ph  ->  E. x A. y ps )
4 vex 2804 . . . . . . . 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 3017 . . . . . 6  |-  ( (
ph  /\  A. y ps )  ->  [. y  /  y ]. ps )
8 sbcid 3020 . . . . . 6  |-  ( [. y  /  y ]. ps  <->  ps )
97, 8sylib 188 . . . . 5  |-  ( (
ph  /\  A. y ps )  ->  ps )
10 sbcid 3020 . . . . 5  |-  ( [. x  /  x ]. ps  <->  ps )
119, 10sylibr 203 . . . 4  |-  ( (
ph  /\  A. y ps )  ->  [. x  /  x ]. ps )
1211spesbcd 3086 . . 3  |-  ( (
ph  /\  A. y ps )  ->  E. x ps )
131, 2, 3, 12exlimdd 1842 . 2  |-  ( ph  ->  E. x ps )
1413alrimiv 1621 1  |-  ( ph  ->  A. y E. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531    e. wcel 1696   _Vcvv 2801   [.wsbc 3004
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  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005
  Copyright terms: Public domain W3C validator