Theorem mp3an2ani 1496

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 1495	. 2 ⊢ ((𝜓 ∧ (𝜓 ∧ 𝜃)) → 𝜂)
6	5	anabss5 680	1 ⊢ ((𝜓 ∧ 𝜃) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104

This theorem is used by:  01sqrexlem4  15303  coprm  16776  frlmssuvc1  21955  en2top  23153  tgrest  23327  pi1cof  25229  voliunlem1  25720  dvnfre  26122  dvcnvre  26189  ig1pdvds  26348  taylthlem2  26548  chtub  27387  2lgsoddprmlem2  27584  fzo0opth  33159  nsgmgc  33730  omabs2  44087  isosctrlem1ALT  45670  chnsubseqwl  47623  odz2prm2pw  48343  lighneallem4  48390  itcovalpclem2  49479  itcovalt2lem2  49484