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 31010. (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  8604  maxprmfct  16885  chpmat1dlem  23153  chpdmatlem2  23157  leordtvallem1  23528  leordtvallem2  23529  mbfmax  25970  wlklnwwlkln2lem  30471  0wlkonlem1  30709  2cycl2d  35912  relowlpssretop  38287  ntrclsk13  45070  smonoord  48446