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  9168  rankelun  9721  mul02  11246  fnn0ind  12512  supminf  12768  nn0p1elfzo  13523  faclbnd5  14105  pfxccatin12lem3  14535  mulre  14923  divalglem0  16193  algcvga  16373  infpn2  16703  prmgaplem7  16847  blssioo  24056  i1fsub  24971  itg1sub  24972  coesub  25516  dgrsub  25531  sincosq1eq  25767  logtayl2  25915  cxploglim  26225  uspgr2v1e2w  27848  ftc1anclem6  35953  io1ii  46554