Theorem mpanr2 438

Description: An inference based on modus ponens. (Contributed by NM, 3-May-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Hypotheses

Ref	Expression
mpanr2.1	⊢ 𝜒
mpanr2.2	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)

Assertion

Ref	Expression
mpanr2	⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Proof of Theorem mpanr2

Step	Hyp	Ref	Expression
1		mpanr2.1	. . 3 ⊢ 𝜒
2	1	jctr 315	. 2 ⊢ (𝜓 → (𝜓 ∧ 𝜒))
3		mpanr2.2	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
4	2, 3	sylan2 286	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem is referenced by:  op1steq  6275  fpmg  6771  pmresg  6773  pw2f1odc  6944  pm54.43  7310  prarloclemarch2  7545  prarloclemlt  7619  prsradd  7912  muleqadd  8754  divdivap1  8809  addltmul  9287  elfzp1b  10232  elfzm1b  10233  expp1zap  10746  expm1ap  10747  fiinbas  14571  opnneissb  14677  blssec  14960