Theorem mp3an2ani 1497

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 1496	. 2 ⊢ ((𝜓 ∧ (𝜓 ∧ 𝜃)) → 𝜂)
6	5	anabss5 681	1 ⊢ ((𝜓 ∧ 𝜃) → 𝜂)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103

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 402  df-3an 1105

This theorem is used by:  01sqrexlem4  15412  coprm  16887  frlmssuvc1  22100  en2top  23303  tgrest  23477  pi1cof  25380  voliunlem1  25871  dvnfre  26272  dvcnvre  26339  ig1pdvds  26498  taylthlem2  26701  chtub  27539  2lgsoddprmlem2  27736  fzo0opth  33395  nsgmgc  33963  omabs2  44334  isosctrlem1ALT  45916  chnsubseqwl  47889  odz2prm2pw  48648  lighneallem4  48695  itcovalpclem2  49783  itcovalt2lem2  49788