Theorem mpanr2 435

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 313	. 2 ⊢ (𝜓 → (𝜓 ∧ 𝜒))
3		mpanr2.2	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
4	2, 3	sylan2 284	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem is referenced by:  op1steq  6139  fpmg  6631  pmresg  6633  pm54.43  7137  prarloclemarch2  7351  prarloclemlt  7425  prsradd  7718  muleqadd  8556  divdivap1  8610  addltmul  9084  elfzp1b  10022  elfzm1b  10023  expp1zap  10494  expm1ap  10495  fiinbas  12588  opnneissb  12696  blssec  12979