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  7452  enumct  7456  djuen  7568  ndvdssub  12716  sgrpidmndm  13786  conjsubgen  14134  cntzssv  14154  cntzi  14156  xmeteq0  15551  xmettri2  15553  metcnpi  15707  metcnpi2  15708  dvbssntrcntop  15876  upgrm  16512  umgrpredgv  16559