Theorem mpanl1 434

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)

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:  mpanl12  436  ercnv  6648  rec11api  8833  divdiv23apzi  8845  recp1lt1  8979  divgt0i  8990  divge0i  8991  ltreci  8992  lereci  8993  lt2msqi  8994  le2msqi  8995  msq11i  8996  ltdiv23i  9006  fnn0ind  9496  elfzp1b  10226  elfzm1b  10227  sqrt11i  11487  sqrtmuli  11488  sqrtmsq2i  11490  sqrtlei  11491  sqrtlti  11492