Theorem mpanl1 430

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 312	. 2 ⊢ (𝜓 → (𝜑 ∧ 𝜓))
3		mpanl1.2	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
4	2, 3	sylan 281	1 ⊢ ((𝜓 ∧ 𝜒) → 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:  mpanl12  432  ercnv  6450  rec11api  8513  divdiv23apzi  8525  recp1lt1  8657  divgt0i  8668  divge0i  8669  ltreci  8670  lereci  8671  lt2msqi  8672  le2msqi  8673  msq11i  8674  ltdiv23i  8684  fnn0ind  9167  elfzp1b  9877  elfzm1b  9878  sqrt11i  10904  sqrtmuli  10905  sqrtmsq2i  10907  sqrtlei  10908  sqrtlti  10909