Theorem tsim1 33937

Description: A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)

Assertion

Ref	Expression
tsim1	|- ( th -> ( ( -. ph \/ ps ) \/ -. ( ph -> ps ) ) )

Proof of Theorem tsim1

Step	Hyp	Ref	Expression
1		exmid 431	. . 3 |- ( ( ph -> ps ) \/ -. ( ph -> ps ) )
2		df-or 385	. . . . 5 |- ( ( -. ph \/ ps ) <-> ( -. -. ph -> ps ) )
3		notnotb 304	. . . . . . 7 |- ( ph <-> -. -. ph )
4	3	bicomi 214	. . . . . 6 |- ( -. -. ph <-> ph )
5	4	imbi1i 339	. . . . 5 |- ( ( -. -. ph -> ps ) <-> ( ph -> ps ) )
6	2, 5	bitri 264	. . . 4 |- ( ( -. ph \/ ps ) <-> ( ph -> ps ) )
7	6	orbi1i 542	. . 3 |- ( ( ( -. ph \/ ps ) \/ -. ( ph -> ps ) ) <-> ( ( ph -> ps ) \/ -. ( ph -> ps ) ) )
8	1, 7	mpbir 221	. 2 |- ( ( -. ph \/ ps ) \/ -. ( ph -> ps ) )
9	8	a1i 11	1 |- ( th -> ( ( -. ph \/ ps ) \/ -. ( ph -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383

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

This theorem is referenced by:  mpt2bi123f  33971  ac6s6  33980