Theorem mpanr1 429

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

Hypotheses

Ref	Expression
mpanr1.1	⊢ 𝜓
mpanr1.2	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)

Assertion

Ref	Expression
mpanr1	⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Proof of Theorem mpanr1

Step	Hyp	Ref	Expression
1		mpanr1.1	. 2 ⊢ 𝜓
2		mpanr1.2	. . 3 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
3	2	anassrs 393	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
4	1, 3	mpanl2 427	1 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 103

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

This theorem is referenced by:  mpanr12  431  axcnre  7479  rec11api  8283  divdiv23apzi  8295  recp1lt1  8423  divgt0i  8434  divge0i  8435  ltreci  8436  lereci  8437  lt2msqi  8438  le2msqi  8439  msq11i  8440  ltdiv23i  8450  ge0gtmnf  9348  sqrt11i  10628  sqrtmuli  10629  sqrtmsq2i  10631  sqrtlei  10632  sqrtlti  10633  cos01gt0  11116