Description: Theorem 5.8 of Clemente p. 20, 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 | 10;11 | ( ( ps /\ ch ) -> -. th ) | ( ph -> ( ( ps /\ ch ) -> -. th ) ) | Given | $e; adantr to move it into the ND hypothesis |
| 2 | 3;4 | ( ta -> th ) | ( ph -> ( ta -> th ) ) | Given | $e; adantr to move it into the ND hypothesis |
| 3 | 7;8 | ch | ( ph -> ch ) | Given | $e; adantr to move it into the ND hypothesis |
| 4 | 1;2 | ta | ( ph -> ta ) | Given | $e. adantr to move it into the ND hypothesis |
| 5 | 6 | ...| ps | ( ( ph /\ ps ) -> ps ) | ND Hypothesis/Assumption | simpr . New ND hypothesis scope, each reference outside the scope must change antecedent ph to ( ph /\ ps ). |
| 6 | 9 | ... ( ps /\ ch ) | ( ( ph /\ ps ) -> ( ps /\ ch ) ) | /\I 5,3 | jca ( /\I), 6,8 ( adantr to bring in scope) |
| 7 | 5 | ... -. th | ( ( ph /\ ps ) -> -. th ) | ->E 1,6 | mpd ( ->E), 2,4 |
| 8 | 12 | ... th | ( ( ph /\ ps ) -> th ) | ->E 2,4 | mpd ( ->E), 9,11; note the contradiction with ND line 7 (MPE line 5) |
| 9 | 13 | -. ps | ( ph -> -. ps ) | -.I 5,7,8 | pm2.65da ( -.I), 5,12; proof by contradiction. MPE step 6 (ND#5) does not need a reference here, because the assumption is embedded in the antecedents |
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 ; simpr 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, using more of Metamath and MPE's capabilities, is shown in ex-natded5.8-2 .
(Contributed by Mario Carneiro, 9-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ex-natded5.8.1 | |- ( ph -> ( ( ps /\ ch ) -> -. th ) ) |
|
| ex-natded5.8.2 | |- ( ph -> ( ta -> th ) ) |
||
| ex-natded5.8.3 | |- ( ph -> ch ) |
||
| ex-natded5.8.4 | |- ( ph -> ta ) |
||
| Assertion | ex-natded5.8 | |- ( ph -> -. ps ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ex-natded5.8.1 | |- ( ph -> ( ( ps /\ ch ) -> -. th ) ) |
|
| 2 | ex-natded5.8.2 | |- ( ph -> ( ta -> th ) ) |
|
| 3 | ex-natded5.8.3 | |- ( ph -> ch ) |
|
| 4 | ex-natded5.8.4 | |- ( ph -> ta ) |
|
| 5 | 4 | adantr | |- ( ( ph /\ ps ) -> ta ) |
| 6 | 2 | adantr | |- ( ( ph /\ ps ) -> ( ta -> th ) ) |
| 7 | 5 6 | mpd | |- ( ( ph /\ ps ) -> th ) |
| 8 | simpr | |- ( ( ph /\ ps ) -> ps ) |
|
| 9 | 3 | adantr | |- ( ( ph /\ ps ) -> ch ) |
| 10 | 8 9 | jca | |- ( ( ph /\ ps ) -> ( ps /\ ch ) ) |
| 11 | 1 | adantr | |- ( ( ph /\ ps ) -> ( ( ps /\ ch ) -> -. th ) ) |
| 12 | 10 11 | mpd | |- ( ( ph /\ ps ) -> -. th ) |
| 13 | 7 12 | pm2.65da | |- ( ph -> -. ps ) |