Theorem mpan2i 677

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 674	1 ⊢ (𝜑 → (𝜓 → 𝜃))

Syntax hints:   → wi 4   ∧ wa 382

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 197  df-an 383

This theorem is referenced by:  tcwf  8910  cflecard  9277  sqrlem7  14193  setciso  16944  lsmss1  18282  sincosq1lem  24466  pjcompi  28867  mdsl1i  29516  dfon2lem3  32022  dfon2lem7  32026  tan2h  33730  dvasin  33824  ismrc  37787  nnsum4primes4  42202  nnsum4primesprm  42204  nnsum4primesgbe  42206  nnsum4primesle9  42208  rngciso  42507  rngcisoALTV  42519  ringciso  42558  ringcisoALTV  42582  aacllem  43075