Theorem simpl2im 384

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 109	. 2 ⊢ ((𝜓 ∧ 𝜒) → 𝜒)
3		simpl2im.2	. 2 ⊢ (𝜒 → 𝜃)
4	1, 2, 3	3syl 17	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 106

This theorem is referenced by:  ctssdccl  7088  enumct  7092  djuen  7188  ndvdssub  11889  sgrpidmndm  12656  xmeteq0  13153  xmettri2  13155  metcnpi  13309  metcnpi2  13310  dvbssntrcntop  13447