Theorem syl11 31

Description: A syllogism inference. Commuted form of an instance of syl 14. (Contributed by BJ, 25-Oct-2021.)

Hypotheses

Ref	Expression
syl11.1	⊢ (𝜑 → (𝜓 → 𝜒))
syl11.2	⊢ (𝜃 → 𝜑)

Assertion

Ref	Expression
syl11	⊢ (𝜓 → (𝜃 → 𝜒))

Proof of Theorem syl11

Step	Hyp	Ref	Expression
1		syl11.2	. . 3 ⊢ (𝜃 → 𝜑)
2		syl11.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	syl 14	. 2 ⊢ (𝜃 → (𝜓 → 𝜒))
4	3	com12 30	1 ⊢ (𝜓 → (𝜃 → 𝜒))

This proof depends on syntax axioms:   → wi 4

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

This theorem is used by:  ssprsseq  3877  elpr2elpr  3901  elovmporab  6289  elovmporab1w  6290  updjud  7422  pfxccatin12  11505  alzdvds  12621  pcmptcl  13121  fiinopn  15105  cnmptcom  15399  metcnp3  15612  ausgrusgrben  16409  usgredg4  16456