Theorem mpan2i 709

Description: An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)

Hypotheses

Ref	Expression
mpan2i.1	⊢ 𝜒
mpan2i.2	⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))

Assertion

Ref	Expression
mpan2i	⊢ (𝜑 → (𝜓 → 𝜃))

Proof of Theorem mpan2i

Step	Hyp	Ref	Expression
1		mpan2i.1	. . 3 ⊢ 𝜒
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝜒)
3		mpan2i.2	. 2 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))
4	2, 3	mpan2d 706	1 ⊢ (𝜑 → (𝜓 → 𝜃))

Syntax hints:   → wi 4   ∧ wa 383

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 196  df-an 385

This theorem is referenced by:  tcwf  8629  cflecard  8958  sqrlem7  13837  setciso  16564  lsmss1  17902  sincosq1lem  24053  pjcompi  27915  mdsl1i  28564  dfon2lem3  30934  dfon2lem7  30938  tan2h  32571  dvasin  32666  ismrc  36282  nnsum4primes4  40205  nnsum4primesprm  40207  nnsum4primesgbe  40209  nnsum4primesle9  40211  rngciso  41774  rngcisoALTV  41786  ringciso  41825  ringcisoALTV  41849  aacllem  42356