Theorem ex-natded5.5 27267

Description: Theorem 5.5 of [Clemente] p. 18, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. 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	2;3	( ps -> ch )	( ph -> ( ps -> ch ) )	Given	$e; adantr 481 to move it into the ND hypothesis
2	5	-. ch	( ph -> -. ch )	Given	$e; we'll use adantr 481 to move it into the ND hypothesis
3	1	...| ps	( ( ph /\ ps ) -> ps )	ND hypothesis assumption	simpr 477
4	4	... ch	( ( ph /\ ps ) -> ch )	->E 1,3	mpd 15 1,3
5	6	... -. ch	( ( ph /\ ps ) -> -. ch )	IT 2	adantr 481 5
6	7	-. ps	( ph -> -. ps )	/\I 3,4,5	pm2.65da 600 4,6

The original used Latin letters; 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. To add an assumption, the antecedent is modified to include it (typically by using adantr 481; simpr 477 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is mtod 189; a proof without context is shown in mto 188.

(Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ex-natded5.5.1	|- ( ph -> ( ps -> ch ) )
ex-natded5.5.2	|- ( ph -> -. ch )

Assertion

Ref	Expression
ex-natded5.5	|- ( ph -> -. ps )

Proof of Theorem ex-natded5.5

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( ph /\ ps ) -> ps )
2		ex-natded5.5.1	. . . 4 |- ( ph -> ( ps -> ch ) )
3	2	adantr 481	. . 3 |- ( ( ph /\ ps ) -> ( ps -> ch ) )
4	1, 3	mpd 15	. 2 |- ( ( ph /\ ps ) -> ch )
5		ex-natded5.5.2	. . 3 |- ( ph -> -. ch )
6	5	adantr 481	. 2 |- ( ( ph /\ ps ) -> -. ch )
7	4, 6	pm2.65da 600	1 |- ( ph -> -. ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by: (None)