Theorem mp3an2ani 1381

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 1380	. 2 ⊢ ((𝜓 ∧ (𝜓 ∧ 𝜃)) → 𝜂)
6	5	anabss5 580	1 ⊢ ((𝜓 ∧ 𝜃) → 𝜂)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-3an 1007

This theorem is referenced by:  tfr1onlemubacc  6579  tfrcllemubacc  6592  mappsrprg  8124  seqclg  10841  seqfeq4g  10900  wrdexg  11243  plusffng  13599  mulgnngsum  13865  ellspsn  14614  metrest  15420  2lgsoddprmlem2  16028