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Theorem wl-embantd 29903
Description: Deduction version of wl-embant 29901. Generalization of a2i 13, imim12i 57, imim1i 58 and imim2i 14, which can be proved by specializing its hypotheses, and some trivial rearrangements. This theorem clarifies in a more general way, under what conditions a wff may be introduced as a nested antecedent to an antecedent. Note: this theorem is not intuitionistically valid (see embantd 54). (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
wl-embantd.1  |-  ( ph  ->  ps )
wl-embantd.2  |-  ( ph  ->  ( ch  ->  th )
)
Assertion
Ref Expression
wl-embantd  |-  ( ph  ->  ( ( ps  ->  ch )  ->  th )
)

Proof of Theorem wl-embantd
StepHypRef Expression
1 wl-embantd.1 . . 3  |-  ( ph  ->  ps )
21pm2.24d 143 . 2  |-  ( ph  ->  ( -.  ps  ->  th ) )
3 wl-embantd.2 . 2  |-  ( ph  ->  ( ch  ->  th )
)
42, 3jad 162 1  |-  ( ph  ->  ( ( ps  ->  ch )  ->  th )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator