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Theorem ex-natded9.26 21732
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 449. Later statements will have this scope.
37;5,4 ...  ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  ps )  A.E 2,y spsbcd 3176 ( A.E), 5,6. To use it we need a1i 11 and vex 2961. This could be immediately done with 19.21bi 1775, 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 3245 ( E.I), 11. To use it we need sylibr 205, which in turn requires sylib 190 and two uses of sbcid 3179. This could be more immediately done using 19.8a 1763, 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 1913 ( E.E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1630 and nfe1 1748 (MPE# 1,2)
614  A. y E. x ps ( x ,  y )  ( ph  ->  A. y E. x ps )  A.I 5 alrimiv 1642 ( 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 21733.

(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 1630 . . 3  |-  F/ x ph
2 nfe1 1748 . . 3  |-  F/ x E. x ps
3 ex-natded9.26.1 . . 3  |-  ( ph  ->  E. x A. y ps )
4 vex 2961 . . . . . . . 8  |-  y  e. 
_V
54a1i 11 . . . . . . 7  |-  ( (
ph  /\  A. y ps )  ->  y  e. 
_V )
6 simpr 449 . . . . . . 7  |-  ( (
ph  /\  A. y ps )  ->  A. y ps )
75, 6spsbcd 3176 . . . . . 6  |-  ( (
ph  /\  A. y ps )  ->  [. y  /  y ]. ps )
8 sbcid 3179 . . . . . 6  |-  ( [. y  /  y ]. ps  <->  ps )
97, 8sylib 190 . . . . 5  |-  ( (
ph  /\  A. y ps )  ->  ps )
10 sbcid 3179 . . . . 5  |-  ( [. x  /  x ]. ps  <->  ps )
119, 10sylibr 205 . . . 4  |-  ( (
ph  /\  A. y ps )  ->  [. x  /  x ]. ps )
1211spesbcd 3245 . . 3  |-  ( (
ph  /\  A. y ps )  ->  E. x ps )
131, 2, 3, 12exlimdd 1913 . 2  |-  ( ph  ->  E. x ps )
1413alrimiv 1642 1  |-  ( ph  ->  A. y E. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   A.wal 1550   E.wex 1551    e. wcel 1726   _Vcvv 2958   [.wsbc 3163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-sbc 3164
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