Theorem jccir 531

Description: Inference conjoining a consequent of a consequent to the right of the consequent in an implication. See also ex-natded5.3i 30831. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by AV, 20-Aug-2019.)

Hypotheses

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

Assertion

Ref	Expression
jccir	⊢ (𝜑 → (𝜓 ∧ 𝜒))

Proof of Theorem jccir

Step	Hyp	Ref	Expression
1		jccir.1	. 2 ⊢ (𝜑 → 𝜓)
2		jccir.2	. . 3 ⊢ (𝜓 → 𝜒)
3	1, 2	syl 18	. 2 ⊢ (𝜑 → 𝜒)
4	1, 3	jca 521	1 ⊢ (𝜑 → (𝜓 ∧ 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  jccil  532  oelim2  8587  maxprmfct  16790  chpmat1dlem  23042  chpdmatlem2  23046  leordtvallem1  23417  leordtvallem2  23418  mbfmax  25859  wlklnwwlkln2lem  30298  0wlkonlem1  30536  2cycl2d  35670  relowlpssretop  38067  ntrclsk13  44855  smonoord  48172