Theorem elimh 1030

Description: Hypothesis builder for the weak deduction theorem. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 26-Jun-2002.) Revised to use the conditional operator. (Revised by BJ, 30-Sep-2019.)

Hypotheses

Ref	Expression
elimh.1	|- ( (if- ( ch , ph , ps ) <-> ph ) -> ( ch <-> ta ) )
elimh.2	|- ( (if- ( ch , ph , ps ) <-> ps ) -> ( th <-> ta ) )
elimh.3	|- th

Assertion

Ref	Expression
elimh	|- ta

Proof of Theorem elimh

Step	Hyp	Ref	Expression
1		ifptru 1023	. . . 4 |- ( ch -> (if- ( ch , ph , ps ) <-> ph ) )
2		elimh.1	. . . 4 |- ( (if- ( ch , ph , ps ) <-> ph ) -> ( ch <-> ta ) )
3	1, 2	syl 17	. . 3 |- ( ch -> ( ch <-> ta ) )
4	3	ibi 256	. 2 |- ( ch -> ta )
5		elimh.3	. . 3 |- th
6		ifpfal 1024	. . . 4 |- ( -. ch -> (if- ( ch , ph , ps ) <-> ps ) )
7		elimh.2	. . . 4 |- ( (if- ( ch , ph , ps ) <-> ps ) -> ( th <-> ta ) )
8	6, 7	syl 17	. . 3 |- ( -. ch -> ( th <-> ta ) )
9	5, 8	mpbii 223	. 2 |- ( -. ch -> ta )
10	4, 9	pm2.61i 176	1 |- ta

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196  if-wif 1012

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 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  con3ALT  1032