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  8196  rec11api  9027  divdiv23apzi  9039  recp1lt1  9173  divgt0i  9184  divge0i  9185  ltreci  9186  lereci  9187  lt2msqi  9188  le2msqi  9189  msq11i  9190  ltdiv23i  9200  ge0gtmnf  10156  sqrt11i  11817  sqrtmuli  11818  sqrtmsq2i  11820  sqrtlei  11821  sqrtlti  11822  cos01gt0  12449