Theorem mp3an3an 1469

Description: mp3an 1463 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 1450	. 2 ⊢ ((𝜒 ∧ 𝜏) → 𝜂)
6	1, 2, 5	syl2an 596	1 ⊢ ((𝜓 ∧ 𝜃) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 395   ∧ 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 207  df-an 396  df-3an 1088

This theorem is referenced by:  mp3an2ani  1470  unfilem2  9190  rankelun  9765  mul02  11291  fnn0ind  12572  supminf  12833  nn0p1elfzo  13602  faclbnd5  14205  pfxccatin12lem3  14639  mulre  15028  divalglem0  16304  algcvga  16490  infpn2  16825  prmgaplem7  16969  blssioo  24710  i1fsub  25636  itg1sub  25637  coesub  26189  dgrsub  26205  sincosq1eq  26448  logtayl2  26598  cxploglim  26915  uspgr2v1e2w  29229  ftc1anclem6  37737  plusmod5ne  47444  io1ii  49020