Theorem tsbi2 33941

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

Assertion

Ref	Expression
tsbi2	|- ( th -> ( ( ph \/ ps ) \/ ( ph <-> ps ) ) )

Proof of Theorem tsbi2

Step	Hyp	Ref	Expression
1		pm5.21 903	. . . 4 |- ( ( -. ph /\ -. ps ) -> ( ph <-> ps ) )
2	1	olcd 408	. . 3 |- ( ( -. ph /\ -. ps ) -> ( ( ph \/ ps ) \/ ( ph <-> ps ) ) )
3		pm4.57 518	. . . . 5 |- ( -. ( -. ph /\ -. ps ) <-> ( ph \/ ps ) )
4	3	biimpi 206	. . . 4 |- ( -. ( -. ph /\ -. ps ) -> ( ph \/ ps ) )
5	4	orcd 407	. . 3 |- ( -. ( -. ph /\ -. ps ) -> ( ( ph \/ ps ) \/ ( ph <-> ps ) ) )
6	2, 5	pm2.61i 176	. 2 |- ( ( ph \/ ps ) \/ ( ph <-> ps ) )
7	6	a1i 11	1 |- ( th -> ( ( ph \/ ps ) \/ ( ph <-> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384

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

This theorem is referenced by:  tsxo2  33945  mpt2bi123f  33971  mptbi12f  33975  ac6s6  33980