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 Translation	ND Rationale	MPE Rationale
1	15	( ps \/ ch )	( ph -> ( ps \/ ch ) )	Given	$e.
2;3	2	( ps -> th )	( ph -> ( ps -> th ) )	Given	$e. adantr to move it into the ND hypothesis
3	9	( -. ta -> -. ch )	( ph -> ( -. ta -> -. ch ) )	Given	$e. ad2antrr to move it into the ND sub-hypothesis
4	1	...\| ps	( ( ph /\ ps ) -> ps )	ND hypothesis assumption	simpr
5	4	... th	( ( ph /\ ps ) -> th )	->E 2,4	mpd 1,3
6	5	... ( th \/ ta )	( ( ph /\ ps ) -> ( th \/ ta ) )	\/I 5	orcd 4
7	6	...\| ch	( ( ph /\ ch ) -> ch )	ND hypothesis assumption	simpr
8	8	... ...\| -. ta	( ( ( ph /\ ch ) /\ -. ta ) -> -. ta )	(sub) ND hypothesis assumption	simpr
9	11	... ... -. ch	( ( ( ph /\ ch ) /\ -. ta ) -> -. ch )	->E 3,8	mpd 8,10
10	7	... ... ch	( ( ( ph /\ ch ) /\ -. ta ) -> ch )	IT 7	adantr 6
11	12	... -. -. ta	( ( ph /\ ch ) -> -. -. ta )	-.I 8,9,10	pm2.65da 7,11
12	13	... ta	( ( ph /\ ch ) -> ta )	-.E 11	notnotrd 12
13	14	... ( th \/ ta )	( ( ph /\ ch ) -> ( th \/ ta ) )	\/I 12	olcd 13
14	16	( 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	$⊢ φ \to (ψ \lor χ)$
	ex-natded5.13.2	$⊢ φ \to (ψ \to θ)$
	ex-natded5.13.3	$⊢ φ \to (\neg τ \to \neg χ)$
Assertion	ex-natded5.13	$⊢ φ \to (θ \lor τ)$

Proof

Step	Hyp	Ref	Expression
1		ex-natded5.13.1	$⊢ φ \to (ψ \lor χ)$
2		ex-natded5.13.2	$⊢ φ \to (ψ \to θ)$
3		ex-natded5.13.3	$⊢ φ \to (\neg τ \to \neg χ)$
4		simpr	$⊢ (φ \land ψ) \to ψ$
5	2	adantr	$⊢ (φ \land ψ) \to (ψ \to θ)$
6	4 5	mpd	$⊢ (φ \land ψ) \to θ$
7	6	orcd	$⊢ (φ \land ψ) \to (θ \lor τ)$
8		simpr	$⊢ (φ \land χ) \to χ$
9	8	adantr	$⊢ ((φ \land χ) \land \neg τ) \to χ$
10		simpr	$⊢ ((φ \land χ) \land \neg τ) \to \neg τ$
11	3	ad2antrr	$⊢ ((φ \land χ) \land \neg τ) \to (\neg τ \to \neg χ)$
12	10 11	mpd	$⊢ ((φ \land χ) \land \neg τ) \to \neg χ$
13	9 12	pm2.65da	$⊢ (φ \land χ) \to \neg \neg τ$
14	13	notnotrd	$⊢ (φ \land χ) \to τ$
15	14	olcd	$⊢ (φ \land χ) \to (θ \lor τ)$
16	7 15 1	mpjaodan	$⊢ φ \to (θ \lor τ)$