Theorem jccil 561

Description: Inference conjoining a consequent of a consequent to the left of the consequent in an implication. Remark: One can also prove this theorem using syl 17 and jca 553 (as done in jccir 560), which would be 4 bytes shorter, but one step longer than the current proof. (Proof modification is discouraged.) (Contributed by AV, 20-Aug-2019.)

Hypotheses

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

Assertion

Ref	Expression
jccil	⊢ (𝜑 → (𝜒 ∧ 𝜓))

Proof of Theorem jccil

Step	Hyp	Ref	Expression
1		jccir.1	. . 3 ⊢ (𝜑 → 𝜓)
2		jccir.2	. . 3 ⊢ (𝜓 → 𝜒)
3	1, 2	jccir 560	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒))
4	3	ancomd 466	1 ⊢ (𝜑 → (𝜒 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 196  df-an 385

This theorem is referenced by:  inatsk  9479  relexpindlem  13651  el2wlksoton  26405  el2spthsoton  26406