Theorem an42ds

Description: Inference exchanging the last antecedent with the second one. See also an32s . (Contributed by Thierry Arnoux, 3-Jun-2025)

	Ref	Expression
Hypothesis	an42ds.1	|- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta )
Assertion	an42ds	|- ( ( ( ( ph /\ th ) /\ ch ) /\ ps ) -> ta )

Proof

Step	Hyp	Ref	Expression
1		an42ds.1	|- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) -> ta )
2		an32	|- ( ( ( ph /\ ps ) /\ th ) <-> ( ( ph /\ th ) /\ ps ) )
3	2	anbi1i	|- ( ( ( ( ph /\ ps ) /\ th ) /\ ch ) <-> ( ( ( ph /\ th ) /\ ps ) /\ ch ) )
4		an32	|- ( ( ( ( ph /\ ps ) /\ th ) /\ ch ) <-> ( ( ( ph /\ ps ) /\ ch ) /\ th ) )
5		an32	|- ( ( ( ( ph /\ th ) /\ ps ) /\ ch ) <-> ( ( ( ph /\ th ) /\ ch ) /\ ps ) )
6	3 4 5	3bitr3i	|- ( ( ( ( ph /\ ps ) /\ ch ) /\ th ) <-> ( ( ( ph /\ th ) /\ ch ) /\ ps ) )
7	6 1	sylbir	|- ( ( ( ( ph /\ th ) /\ ch ) /\ ps ) -> ta )