Theorem mpanl2 439

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 315	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝜓))
3		mpanl2.2	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)
4	2, 3	sylan 283	1 ⊢ ((𝜑 ∧ 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:  mpanr1  441  mp3an2  1366  reuss  3514  tfri3  6638  prarloclemarch2  7786  prarloclemlt  7860  prsradd  8153  map2psrprg  8172  pitonnlem2  8214  axcnre  8248  muleqadd  8999  divdivap2  9055  addltmul  9544  clwwlknonex2lem2  16691