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Theorem ex-natded5.3 20810
Description: Theorem 5.3 of [Clemente] p. 16, translated line by line using an interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.3-2 20811. A proof without context is shown in ex-natded5.3i 20812. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3  ( ps  ->  ch )  ( ph  ->  ( ps  ->  ch ) ) Given $e; adantr 451 to move it into the ND hypothesis
25;6  ( ch  ->  th )  ( ph  ->  ( ch  ->  th ) ) Given $e; adantr 451 to move it into the ND hypothesis
31 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 447, to access the new assumption
44 ...  ch  ( ( ph  /\  ps )  ->  ch )  ->E 1,3 mpd 14, the MPE equivalent of  ->E, 1.3. adantr 451 was used to transform its dependency (we could also use imp 418 to get this directly from 1)
57 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 14, the MPE equivalent of  ->E, 4,6. adantr 451 was used to transform its dependency
68 ...  ( ch  /\  th )  ( ( ph  /\  ps )  ->  ( ch  /\  th ) )  /\I 4,5 jca 518, the MPE equivalent of  /\I, 4,7
79  ( ps  ->  ( ch  /\  th ) )  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )  ->I 3,6 ex 423, the MPE equivalent of  ->I, 8

The original used Latin letters for predicates; 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. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypotheses
Ref Expression
ex-natded5.3.1  |-  ( ph  ->  ( ps  ->  ch ) )
ex-natded5.3.2  |-  ( ph  ->  ( ch  ->  th )
)
Assertion
Ref Expression
ex-natded5.3  |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )

Proof of Theorem ex-natded5.3
StepHypRef Expression
1 simpr 447 . . . 4  |-  ( (
ph  /\  ps )  ->  ps )
2 ex-natded5.3.1 . . . . 5  |-  ( ph  ->  ( ps  ->  ch ) )
32adantr 451 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ps  ->  ch ) )
41, 3mpd 14 . . 3  |-  ( (
ph  /\  ps )  ->  ch )
5 ex-natded5.3.2 . . . . 5  |-  ( ph  ->  ( ch  ->  th )
)
65adantr 451 . . . 4  |-  ( (
ph  /\  ps )  ->  ( ch  ->  th )
)
74, 6mpd 14 . . 3  |-  ( (
ph  /\  ps )  ->  th )
84, 7jca 518 . 2  |-  ( (
ph  /\  ps )  ->  ( ch  /\  th ) )
98ex 423 1  |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360
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