Theorem syl2anc2 416

Description: Double syllogism inference combined with contraction. (Contributed by BTernaryTau, 29-Sep-2023.)

Hypotheses

Ref	Expression
syl2anc2.1	|- ( ph -> ps )
syl2anc2.2	|- ( ps -> ch )
syl2anc2.3	|- ( ( ps /\ ch ) -> th )

Assertion

Ref	Expression
syl2anc2	|- ( ph -> th )

Proof of Theorem syl2anc2

Step	Hyp	Ref	Expression
1		syl2anc2.1	. 2 |- ( ph -> ps )
2		syl2anc2.2	. . 3 |- ( ps -> ch )
3	1, 2	syl 14	. 2 |- ( ph -> ch )
4		syl2anc2.3	. 2 |- ( ( ps /\ ch ) -> th )
5	1, 3, 4	syl2anc 415	1 |- ( ph -> th )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108

This theorem is used by:  0mhm  13793  grpidssd  13881  gsumzfi  14158  gsummptfidmadd  14161  rnglz  14244  rngrz  14245  ringlz  14348  ringrz  14349  issubrng2  14518  issubrg2  14549