Theorem ex-natded5.13 28779

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 28780. 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 481 to move it into the ND hypothesis
3	9	(¬ 𝜏 → ¬ 𝜒)	(𝜑 → (¬ 𝜏 → ¬ 𝜒))	Given	$e. ad2antrr 723 to move it into the ND sub-hypothesis
4	1	...| 𝜓	((𝜑 ∧ 𝜓) → 𝜓)	ND hypothesis assumption	simpr 485
5	4	... 𝜃	((𝜑 ∧ 𝜓) → 𝜃)	→E 2,4	mpd 15 1,3
6	5	... (𝜃 ∨ 𝜏)	((𝜑 ∧ 𝜓) → (𝜃 ∨ 𝜏))	∨I 5	orcd 870 4
7	6	...| 𝜒	((𝜑 ∧ 𝜒) → 𝜒)	ND hypothesis assumption	simpr 485
8	8	... ...| ¬ 𝜏	(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜏)	(sub) ND hypothesis assumption	simpr 485
9	11	... ... ¬ 𝜒	(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜒)	→E 3,8	mpd 15 8,10
10	7	... ... 𝜒	(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → 𝜒)	IT 7	adantr 481 6
11	12	... ¬ ¬ 𝜏	((𝜑 ∧ 𝜒) → ¬ ¬ 𝜏)	¬I 8,9,10	pm2.65da 814 7,11
12	13	... 𝜏	((𝜑 ∧ 𝜒) → 𝜏)	¬E 11	notnotrd 133 12
13	14	... (𝜃 ∨ 𝜏)	((𝜑 ∧ 𝜒) → (𝜃 ∨ 𝜏))	∨I 12	olcd 871 13
14	16	(𝜃 ∨ 𝜏)	(𝜑 → (𝜃 ∨ 𝜏))	∨E 1,6,13	mpjaodan 956 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 481; simpr 485 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 485	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
2		ex-natded5.13.2	. . . . 5 ⊢ (𝜑 → (𝜓 → 𝜃))
3	2	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃))
4	1, 3	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
5	4	orcd 870	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 ∨ 𝜏))
6		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝜒) → 𝜒)
7	6	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → 𝜒)
8		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜏)
9		ex-natded5.13.3	. . . . . . 7 ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒))
10	9	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → (¬ 𝜏 → ¬ 𝜒))
11	8, 10	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜒)
12	7, 11	pm2.65da 814	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → ¬ ¬ 𝜏)
13	12	notnotrd 133	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜏)
14	13	olcd 871	. 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜃 ∨ 𝜏))
15		ex-natded5.13.1	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
16	5, 14, 15	mpjaodan 956	1 ⊢ (𝜑 → (𝜃 ∨ 𝜏))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844

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 397  df-or 845

This theorem is referenced by: (None)