Theorem nanbi12 1457

Description: Join two logical equivalences with anti-conjunction. (Contributed by SF, 2-Jan-2018.)

Assertion

Ref	Expression
nanbi12	|- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ph -/\ ch ) <-> ( ps -/\ th ) ) )

Proof of Theorem nanbi12

Step	Hyp	Ref	Expression
1		nanbi1 1455	. 2 |- ( ( ph <-> ps ) -> ( ( ph -/\ ch ) <-> ( ps -/\ ch ) ) )
2		nanbi2 1456	. 2 |- ( ( ch <-> th ) -> ( ( ps -/\ ch ) <-> ( ps -/\ th ) ) )
3	1, 2	sylan9bb 736	1 |- ( ( ( ph <-> ps ) /\ ( ch <-> th ) ) -> ( ( ph -/\ ch ) <-> ( ps -/\ th ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   -/\ wnan 1447

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-an 386  df-nan 1448

This theorem is referenced by:  nanbi12i  1460  nanbi12d  1463