Theorem mp3an3an 1466

Description: mp3an 1460 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 1447	. 2 ⊢ ((𝜒 ∧ 𝜏) → 𝜂)
6	1, 2, 5	syl2an 596	1 ⊢ ((𝜓 ∧ 𝜃) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086

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 397  df-3an 1088

This theorem is referenced by:  mp3an2ani  1467  unfilem2  9079  rankelun  9630  mul02  11153  fnn0ind  12419  supminf  12675  nn0p1elfzo  13430  faclbnd5  14012  pfxccatin12lem3  14445  mulre  14832  divalglem0  16102  algcvga  16284  infpn2  16614  prmgaplem7  16758  blssioo  23958  i1fsub  24873  itg1sub  24874  coesub  25418  dgrsub  25433  sincosq1eq  25669  logtayl2  25817  cxploglim  26127  uspgr2v1e2w  27618  ftc1anclem6  35855  io1ii  46214