# Metamath Proof Explorer

## Theorem ex-natded5.13

Description: Theorem 5.13 of Clemente p. 20, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.13-2 . 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
115 ( ps \/ ch ) ( ph -> ( ps \/ ch ) ) Given $e. 2;32 ( ps -> th ) ( ph -> ( ps -> th ) ) Given$e. adantr to move it into the ND hypothesis
39 ( -. ta -> -. ch ) ( ph -> ( -. ta -> -. ch ) ) Given \$e. ad2antrr to move it into the ND sub-hypothesis
41 ...| ps ( ( ph /\ ps ) -> ps ) ND hypothesis assumption simpr
54 ... th ( ( ph /\ ps ) -> th ) ->E 2,4 mpd 1,3
65 ... ( th \/ ta ) ( ( ph /\ ps ) -> ( th \/ ta ) ) \/I 5 orcd 4
76 ...| ch ( ( ph /\ ch ) -> ch ) ND hypothesis assumption simpr
88 ... ...| -. ta ( ( ( ph /\ ch ) /\ -. ta ) -> -. ta ) (sub) ND hypothesis assumption simpr
911 ... ... -. ch ( ( ( ph /\ ch ) /\ -. ta ) -> -. ch ) ->E 3,8 mpd 8,10
107 ... ... ch ( ( ( ph /\ ch ) /\ -. ta ) -> ch ) IT 7 adantr 6
1112 ... -. -. ta ( ( ph /\ ch ) -> -. -. ta ) -.I 8,9,10 pm2.65da 7,11
1213 ... ta ( ( ph /\ ch ) -> ta ) -.E 11 notnotrd 12
1314 ... ( th \/ ta ) ( ( ph /\ ch ) -> ( th \/ ta ) ) \/I 12 olcd 13
1416 ( th \/ ta ) ( ph -> ( th \/ ta ) ) \/E 1,6,13 mpjaodan 5,14,15

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). (Contributed by Mario Carneiro, 9-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ex-natded5.13.1 ${⊢}{\phi }\to \left({\psi }\vee {\chi }\right)$
ex-natded5.13.2 ${⊢}{\phi }\to \left({\psi }\to {\theta }\right)$
ex-natded5.13.3 ${⊢}{\phi }\to \left(¬{\tau }\to ¬{\chi }\right)$
Assertion ex-natded5.13 ${⊢}{\phi }\to \left({\theta }\vee {\tau }\right)$

### Proof

Step Hyp Ref Expression
1 ex-natded5.13.1 ${⊢}{\phi }\to \left({\psi }\vee {\chi }\right)$
2 ex-natded5.13.2 ${⊢}{\phi }\to \left({\psi }\to {\theta }\right)$
3 ex-natded5.13.3 ${⊢}{\phi }\to \left(¬{\tau }\to ¬{\chi }\right)$
4 simpr ${⊢}\left({\phi }\wedge {\psi }\right)\to {\psi }$
5 2 adantr ${⊢}\left({\phi }\wedge {\psi }\right)\to \left({\psi }\to {\theta }\right)$
6 4 5 mpd ${⊢}\left({\phi }\wedge {\psi }\right)\to {\theta }$
7 6 orcd ${⊢}\left({\phi }\wedge {\psi }\right)\to \left({\theta }\vee {\tau }\right)$
8 simpr ${⊢}\left({\phi }\wedge {\chi }\right)\to {\chi }$
9 8 adantr ${⊢}\left(\left({\phi }\wedge {\chi }\right)\wedge ¬{\tau }\right)\to {\chi }$
10 simpr ${⊢}\left(\left({\phi }\wedge {\chi }\right)\wedge ¬{\tau }\right)\to ¬{\tau }$
11 3 ad2antrr ${⊢}\left(\left({\phi }\wedge {\chi }\right)\wedge ¬{\tau }\right)\to \left(¬{\tau }\to ¬{\chi }\right)$
12 10 11 mpd ${⊢}\left(\left({\phi }\wedge {\chi }\right)\wedge ¬{\tau }\right)\to ¬{\chi }$
13 9 12 pm2.65da ${⊢}\left({\phi }\wedge {\chi }\right)\to ¬¬{\tau }$
14 13 notnotrd ${⊢}\left({\phi }\wedge {\chi }\right)\to {\tau }$
15 14 olcd ${⊢}\left({\phi }\wedge {\chi }\right)\to \left({\theta }\vee {\tau }\right)$
16 7 15 1 mpjaodan ${⊢}{\phi }\to \left({\theta }\vee {\tau }\right)$