Theorem mpanl2 697

Description: An inference based on modus ponens. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Hypotheses

Ref	Expression
mpanl2.1	⊢ 𝜓
mpanl2.2	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Assertion

Ref	Expression
mpanl2	⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Proof of Theorem mpanl2

Step	Hyp	Ref	Expression
1		mpanl2.1	. . 3 ⊢ 𝜓
2	1	jctr 523	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝜓))
3		mpanl2.2	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
4	2, 3	sylan 578	1 ⊢ ((𝜑 ∧ 𝜒) → 𝜃)

Syntax hints:   → wi 4   ∧ wa 394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 395

This theorem is referenced by:  mpanr1  699  mp3an2  1447  reuss  4315  tfrlem11  8390  tfr3  8401  oe0  8524  unfi  9174  dif1ennnALT  9279  indpi  10904  map2psrpr  11107  axcnre  11161  muleqadd  11862  divdiv2  11930  addltmul  12452  supxrpnf  13301  supxrunb1  13302  supxrunb2  13303  iimulcl  24680  clwwlknonex2lem2  29628  nmopadjlem  31609  nmopcoadji  31621  opsqrlem6  31665  hstrbi  31786  sgncl  33835  poimirlem3  36794  dflim5  42381  aacllem  47935