Theorem tsbi3 33942

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

Assertion

Ref	Expression
tsbi3	|- ( th -> ( ( ph \/ -. ps ) \/ -. ( ph <-> ps ) ) )

Proof of Theorem tsbi3

Step	Hyp	Ref	Expression
1		biimpr 210	. . . . 5 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
2		con34b 306	. . . . . 6 |- ( ( ps -> ph ) <-> ( -. ph -> -. ps ) )
3		pm2.54 389	. . . . . 6 |- ( ( -. ph -> -. ps ) -> ( ph \/ -. ps ) )
4	2, 3	sylbi 207	. . . . 5 |- ( ( ps -> ph ) -> ( ph \/ -. ps ) )
5	1, 4	syl 17	. . . 4 |- ( ( ph <-> ps ) -> ( ph \/ -. ps ) )
6	5	con3i 150	. . 3 |- ( -. ( ph \/ -. ps ) -> -. ( ph <-> ps ) )
7	6	orri 391	. 2 |- ( ( ph \/ -. ps ) \/ -. ( ph <-> ps ) )
8	7	a1i 11	1 |- ( th -> ( ( ph \/ -. ps ) \/ -. ( ph <-> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ 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:  tsbi4  33943  tsxo3  33946  mpt2bi123f  33971  mptbi12f  33975  ac6s6  33980