Theorem mp3an3an 1470

Description: mp3an 1464 with antecedents in standard conjunction form and with two hypotheses which are implications. (Contributed by Alan Sare, 28-Aug-2016.)

Hypotheses

Ref	Expression
mp3an3an.1	⊢ 𝜑
mp3an3an.2	⊢ (𝜓 → 𝜒)
mp3an3an.3	⊢ (𝜃 → 𝜏)
mp3an3an.4	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
mp3an3an	⊢ ((𝜓 ∧ 𝜃) → 𝜂)

Proof of Theorem mp3an3an

Step	Hyp	Ref	Expression
1		mp3an3an.2	. 2 ⊢ (𝜓 → 𝜒)
2		mp3an3an.3	. 2 ⊢ (𝜃 → 𝜏)
3		mp3an3an.1	. . 3 ⊢ 𝜑
4		mp3an3an.4	. . 3 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)
5	3, 4	mp3an1 1451	. 2 ⊢ ((𝜒 ∧ 𝜏) → 𝜂)
6	1, 2, 5	syl2an 597	1 ⊢ ((𝜓 ∧ 𝜃) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087

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 207  df-an 396  df-3an 1089

This theorem is referenced by:  mp3an2ani  1471  unfilem2  9220  rankelun  9798  mul02  11325  fnn0ind  12605  supminf  12862  nn0p1elfzo  13632  faclbnd5  14235  pfxccatin12lem3  14669  mulre  15058  divalglem0  16334  algcvga  16520  infpn2  16855  prmgaplem7  16999  blssioo  24756  i1fsub  25682  itg1sub  25683  coesub  26235  dgrsub  26251  sincosq1eq  26494  logtayl2  26644  cxploglim  26961  uspgr2v1e2w  29342  ftc1anclem6  37978  plusmod5ne  47734  io1ii  49309