Theorem ex-natded5.13 30212

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 30213. 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	(𝜓 ∨ 𝜒)	(𝜑 → (𝜓 ∨ 𝜒))	Given	$e.
2;3	2	(𝜓 → 𝜃)	(𝜑 → (𝜓 → 𝜃))	Given	$e. adantr 480 to move it into the ND hypothesis
3	9	(¬ 𝜏 → ¬ 𝜒)	(𝜑 → (¬ 𝜏 → ¬ 𝜒))	Given	$e. ad2antrr 725 to move it into the ND sub-hypothesis
4	1	...| 𝜓	((𝜑 ∧ 𝜓) → 𝜓)	ND hypothesis assumption	simpr 484
5	4	... 𝜃	((𝜑 ∧ 𝜓) → 𝜃)	→E 2,4	mpd 15 1,3
6	5	... (𝜃 ∨ 𝜏)	((𝜑 ∧ 𝜓) → (𝜃 ∨ 𝜏))	∨I 5	orcd 872 4
7	6	...| 𝜒	((𝜑 ∧ 𝜒) → 𝜒)	ND hypothesis assumption	simpr 484
8	8	... ...| ¬ 𝜏	(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜏)	(sub) ND hypothesis assumption	simpr 484
9	11	... ... ¬ 𝜒	(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜒)	→E 3,8	mpd 15 8,10
10	7	... ... 𝜒	(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → 𝜒)	IT 7	adantr 480 6
11	12	... ¬ ¬ 𝜏	((𝜑 ∧ 𝜒) → ¬ ¬ 𝜏)	¬I 8,9,10	pm2.65da 816 7,11
12	13	... 𝜏	((𝜑 ∧ 𝜒) → 𝜏)	¬E 11	notnotrd 133 12
13	14	... (𝜃 ∨ 𝜏)	((𝜑 ∧ 𝜒) → (𝜃 ∨ 𝜏))	∨I 12	olcd 873 13
14	16	(𝜃 ∨ 𝜏)	(𝜑 → (𝜃 ∨ 𝜏))	∨E 1,6,13	mpjaodan 957 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 𝜑 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 480; simpr 484 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.)

Hypotheses

Ref	Expression
ex-natded5.13.1	⊢ (𝜑 → (𝜓 ∨ 𝜒))
ex-natded5.13.2	⊢ (𝜑 → (𝜓 → 𝜃))
ex-natded5.13.3	⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒))

Assertion

Ref	Expression
ex-natded5.13	⊢ (𝜑 → (𝜃 ∨ 𝜏))

Proof of Theorem ex-natded5.13

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
2		ex-natded5.13.2	. . . . 5 ⊢ (𝜑 → (𝜓 → 𝜃))
3	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃))
4	1, 3	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
5	4	orcd 872	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 ∨ 𝜏))
6		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝜒) → 𝜒)
7	6	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → 𝜒)
8		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜏)
9		ex-natded5.13.3	. . . . . . 7 ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒))
10	9	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → (¬ 𝜏 → ¬ 𝜒))
11	8, 10	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜒)
12	7, 11	pm2.65da 816	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → ¬ ¬ 𝜏)
13	12	notnotrd 133	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜏)
14	13	olcd 873	. 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜃 ∨ 𝜏))
15		ex-natded5.13.1	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
16	5, 14, 15	mpjaodan 957	1 ⊢ (𝜑 → (𝜃 ∨ 𝜏))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846

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 206  df-an 396  df-or 847

This theorem is referenced by: (None)