Theorem ex-natded9.26 27276

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 Translation	ND Rationale	MPE Rationale
1	3	E. x A. y ps ( x , y )	( ph -> E. x A. y ps )	Given	$e.
2	6	...| A. y ps ( x , y )	( ( ph /\ A. y ps ) -> A. y ps )	ND hypothesis assumption	simpr 477. Later statements will have this scope.
3	7;5,4	... ps ( x , y )	( ( ph /\ A. y ps ) -> ps )	A.E 2,y	spsbcd 3449 ( A.E), 5,6. To use it we need a1i 11 and vex 3203. This could be immediately done with 19.21bi 2059, but we want to show the general approach for substitution.
4	12;8,9,10,11	... E. x ps ( x , y )	( ( ph /\ A. y ps ) -> E. x ps )	E.I 3,a	spesbcd 3522 ( E.I), 11. To use it we need sylibr 224, which in turn requires sylib 208 and two uses of sbcid 3452. This could be more immediately done using 19.8a 2052, but we want to show the general approach for substitution.
5	13;1,2	E. x ps ( x , y )	( ph -> E. x ps )	E.E 1,2,4,a	exlimdd 2088 ( E.E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1843 and nfe1 2027 (MPE# 1,2)
6	14	A. y E. x ps ( x , y )	( ph -> A. y E. x ps )	A.I 5	alrimiv 1855 ( 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 27277.

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

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

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 |- F/ x ph
2		nfe1 2027	. . 3 |- F/ x E. x ps
3		ex-natded9.26.1	. . 3 |- ( ph -> E. x A. y ps )
4		vex 3203	. . . . . . . 8 |- y e. _V
5	4	a1i 11	. . . . . . 7 |- ( ( ph /\ A. y ps ) -> y e. _V )
6		simpr 477	. . . . . . 7 |- ( ( ph /\ A. y ps ) -> A. y ps )
7	5, 6	spsbcd 3449	. . . . . 6 |- ( ( ph /\ A. y ps ) -> [. y / y ]. ps )
8		sbcid 3452	. . . . . 6 |- ( [. y / y ]. ps <-> ps )
9	7, 8	sylib 208	. . . . 5 |- ( ( ph /\ A. y ps ) -> ps )
10		sbcid 3452	. . . . 5 |- ( [. x / x ]. ps <-> ps )
11	9, 10	sylibr 224	. . . 4 |- ( ( ph /\ A. y ps ) -> [. x / x ]. ps )
12	11	spesbcd 3522	. . 3 |- ( ( ph /\ A. y ps ) -> E. x ps )
13	1, 2, 3, 12	exlimdd 2088	. 2 |- ( ph -> E. x ps )
14	13	alrimiv 1855	1 |- ( ph -> A. y E. x ps )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   _Vcvv 3200   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436

This theorem is referenced by: (None)