Theorem mpanl1 438

Description: An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Hypotheses

Ref	Expression
mpanl1.1	⊢ 𝜑
mpanl1.2	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
mpanl1	⊢ ((𝜓 ∧ 𝜒) → 𝜃)

Proof of Theorem mpanl1

Step	Hyp	Ref	Expression
1		mpanl1.1	. . 3 ⊢ 𝜑
2	1	jctl 314	. 2 ⊢ (𝜓 → (𝜑 ∧ 𝜓))
3		mpanl1.2	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
4	2, 3	sylan 283	1 ⊢ ((𝜓 ∧ 𝜒) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem is used by:  mpanl12  440  ercnv  6828  rec11api  9084  divdiv23apzi  9096  recp1lt1  9230  divgt0i  9241  divge0i  9242  ltreci  9243  lereci  9244  lt2msqi  9245  le2msqi  9246  msq11i  9247  ltdiv23i  9257  fnn0ind  9764  elfzp1b  10506  elfzm1b  10507  sqrt11i  11900  sqrtmuli  11901  sqrtmsq2i  11903  sqrtlei  11904  sqrtlti  11905