Theorem tsbi1 33940

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

Assertion

Ref	Expression
tsbi1	|- ( th -> ( ( -. ph \/ -. ps ) \/ ( ph <-> ps ) ) )

Proof of Theorem tsbi1

Step	Hyp	Ref	Expression
1		pm5.1 902	. . . 4 |- ( ( ph /\ ps ) -> ( ph <-> ps ) )
2	1	olcd 408	. . 3 |- ( ( ph /\ ps ) -> ( ( -. ph \/ -. ps ) \/ ( ph <-> ps ) ) )
3		pm3.13 522	. . . 4 |- ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) )
4	3	orcd 407	. . 3 |- ( -. ( ph /\ ps ) -> ( ( -. ph \/ -. ps ) \/ ( ph <-> ps ) ) )
5	2, 4	pm2.61i 176	. 2 |- ( ( -. ph \/ -. ps ) \/ ( ph <-> ps ) )
6	5	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:  tsxo1  33944  mpt2bi123f  33971  mptbi12f  33975  ac6s6  33980