Theorem simpl2im 381

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-1 5  ax-2 6  ax-mp 7  ax-ia2 106

This theorem is referenced by:  ctssdclemr  6911  enumct  6914  djuen  6971  ndvdssub  11422  xmeteq0  12287  xmettri2  12289  metcnpi  12439  metcnpi2  12440  dvbssntrcntop  12526