Theorem simpl2im 390

Description: Implication from an eliminated conjunct implied by the antecedent. (Contributed by BJ/AV, 5-Apr-2021.)

Hypotheses

Ref	Expression
simpl2im.1	⊢ (𝜑 → (𝜓 ∧ 𝜒))
simpl2im.2	⊢ (𝜒 → 𝜃)

Assertion

Ref	Expression
simpl2im	⊢ (𝜑 → 𝜃)

Proof of Theorem simpl2im

Step	Hyp	Ref	Expression
1		simpl2im.1	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒))
2		simpr 110	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
3		simpl2im.2	. 2 ⊢ (𝜒 → 𝜃)
4	1, 2, 3	3syl 17	1 ⊢ (𝜑 → 𝜃)

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-ia2 107

This theorem is used by:  rabsnif  3778  ctssdccl  7451  enumct  7455  djuen  7567  ndvdssub  12699  sgrpidmndm  13735  conjsubgen  14083  xmeteq0  15462  xmettri2  15464  metcnpi  15618  metcnpi2  15619  dvbssntrcntop  15787  upgrm  16353  umgrpredgv  16400