Theorem mp3an2ani 1467

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 1466	. 2 ⊢ ((𝜓 ∧ (𝜓 ∧ 𝜃)) → 𝜂)
6	5	anabss5 668	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:  01sqrexlem4  15281  coprm  16745  frlmssuvc1  21832  en2top  23008  tgrest  23183  pi1cof  25106  voliunlem1  25599  dvnfre  26005  dvcnvre  26073  ig1pdvds  26234  taylthlem2  26431  taylthlem2OLD  26432  chtub  27271  2lgsoddprmlem2  27468  fzo0opth  32813  nsgmgc  33420  omabs2  43322  isosctrlem1ALT  44932  odz2prm2pw  47488  lighneallem4  47535  itcovalpclem2  48521  itcovalt2lem2  48526