Theorem mp3an2ani 1468

Description: An elimination deduction. (Contributed by Alan Sare, 17-Oct-2017.)

Hypotheses

Ref	Expression
mp3an2ani.1	⊢ 𝜑
mp3an2ani.2	⊢ (𝜓 → 𝜒)
mp3an2ani.3	⊢ ((𝜓 ∧ 𝜃) → 𝜏)
mp3an2ani.4	⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
mp3an2ani	⊢ ((𝜓 ∧ 𝜃) → 𝜂)

Proof of Theorem mp3an2ani

Step	Hyp	Ref	Expression
1		mp3an2ani.1	. . 3 ⊢ 𝜑
2		mp3an2ani.2	. . 3 ⊢ (𝜓 → 𝜒)
3		mp3an2ani.3	. . 3 ⊢ ((𝜓 ∧ 𝜃) → 𝜏)
4		mp3an2ani.4	. . 3 ⊢ ((𝜑 ∧ 𝜒 ∧ 𝜏) → 𝜂)
5	1, 2, 3, 4	mp3an3an 1467	. 2 ⊢ ((𝜓 ∧ (𝜓 ∧ 𝜃)) → 𝜂)
6	5	anabss5 666	1 ⊢ ((𝜓 ∧ 𝜃) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 396   ∧ 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 206  df-an 397  df-3an 1089

This theorem is referenced by:  01sqrexlem4  15194  coprm  16650  frlmssuvc1  21355  en2top  22495  tgrest  22670  pi1cof  24582  voliunlem1  25074  dvnfre  25476  dvcnvre  25543  ig1pdvds  25701  taylthlem2  25893  chtub  26722  2lgsoddprmlem2  26919  nsgmgc  32568  omabs2  42164  isosctrlem1ALT  43777  odz2prm2pw  46310  lighneallem4  46357  itcovalpclem2  47435  itcovalt2lem2  47440