Theorem ex-natded5.13 28194

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 28195. 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 483 to move it into the ND hypothesis
3	9	(¬ 𝜏 → ¬ 𝜒)	(𝜑 → (¬ 𝜏 → ¬ 𝜒))	Given	$e. ad2antrr 724 to move it into the ND sub-hypothesis
4	1	...| 𝜓	((𝜑 ∧ 𝜓) → 𝜓)	ND hypothesis assumption	simpr 487
5	4	... 𝜃	((𝜑 ∧ 𝜓) → 𝜃)	→E 2,4	mpd 15 1,3
6	5	... (𝜃 ∨ 𝜏)	((𝜑 ∧ 𝜓) → (𝜃 ∨ 𝜏))	∨I 5	orcd 869 4
7	6	...| 𝜒	((𝜑 ∧ 𝜒) → 𝜒)	ND hypothesis assumption	simpr 487
8	8	... ...| ¬ 𝜏	(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜏)	(sub) ND hypothesis assumption	simpr 487
9	11	... ... ¬ 𝜒	(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜒)	→E 3,8	mpd 15 8,10
10	7	... ... 𝜒	(((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → 𝜒)	IT 7	adantr 483 6
11	12	... ¬ ¬ 𝜏	((𝜑 ∧ 𝜒) → ¬ ¬ 𝜏)	¬I 8,9,10	pm2.65da 815 7,11
12	13	... 𝜏	((𝜑 ∧ 𝜒) → 𝜏)	¬E 11	notnotrd 135 12
13	14	... (𝜃 ∨ 𝜏)	((𝜑 ∧ 𝜒) → (𝜃 ∨ 𝜏))	∨I 12	olcd 870 13
14	16	(𝜃 ∨ 𝜏)	(𝜑 → (𝜃 ∨ 𝜏))	∨E 1,6,13	mpjaodan 955 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 483; simpr 487 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 487	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
2		ex-natded5.13.2	. . . . 5 ⊢ (𝜑 → (𝜓 → 𝜃))
3	2	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃))
4	1, 3	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
5	4	orcd 869	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜃 ∨ 𝜏))
6		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝜒) → 𝜒)
7	6	adantr 483	. . . . 5 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → 𝜒)
8		simpr 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜏)
9		ex-natded5.13.3	. . . . . . 7 ⊢ (𝜑 → (¬ 𝜏 → ¬ 𝜒))
10	9	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → (¬ 𝜏 → ¬ 𝜒))
11	8, 10	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ 𝜒) ∧ ¬ 𝜏) → ¬ 𝜒)
12	7, 11	pm2.65da 815	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → ¬ ¬ 𝜏)
13	12	notnotrd 135	. . 3 ⊢ ((𝜑 ∧ 𝜒) → 𝜏)
14	13	olcd 870	. 2 ⊢ ((𝜑 ∧ 𝜒) → (𝜃 ∨ 𝜏))
15		ex-natded5.13.1	. 2 ⊢ (𝜑 → (𝜓 ∨ 𝜒))
16	5, 14, 15	mpjaodan 955	1 ⊢ (𝜑 → (𝜃 ∨ 𝜏))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843

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 209  df-an 399  df-or 844

This theorem is referenced by: (None)