Theorem mpanr1 437

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 400	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
4	1, 3	mpanl2 435	1 ⊢ ((𝜑 ∧ 𝜒) → 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:  mpanr12  439  axcnre  8144  rec11api  8975  divdiv23apzi  8987  recp1lt1  9121  divgt0i  9132  divge0i  9133  ltreci  9134  lereci  9135  lt2msqi  9136  le2msqi  9137  msq11i  9138  ltdiv23i  9148  ge0gtmnf  10102  sqrt11i  11755  sqrtmuli  11756  sqrtmsq2i  11758  sqrtlei  11759  sqrtlti  11760  cos01gt0  12387