Theorem syl1111anc 854

Description: Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1401 except the unification theorem uses left-nested conjunction. (Contributed by Alan Sare, 17-Oct-2017.)

Hypotheses

Ref	Expression
syl1111anc.1	⊢ (𝜑 → 𝜓)
syl1111anc.2	⊢ (𝜑 → 𝜒)
syl1111anc.3	⊢ (𝜑 → 𝜃)
syl1111anc.4	⊢ (𝜑 → 𝜏)
syl1111anc.5	⊢ ((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
syl1111anc	⊢ (𝜑 → 𝜂)

Proof of Theorem syl1111anc

Step	Hyp	Ref	Expression
1		syl1111anc.1	. . 3 ⊢ (𝜑 → 𝜓)
2		syl1111anc.2	. . 3 ⊢ (𝜑 → 𝜒)
3	1, 2	jca 521	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒))
4		syl1111anc.3	. 2 ⊢ (𝜑 → 𝜃)
5		syl1111anc.4	. 2 ⊢ (𝜑 → 𝜏)
6		syl1111anc.5	. 2 ⊢ ((((𝜓 ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) → 𝜂)
7	3, 4, 5, 6	syl21anc 851	1 ⊢ (𝜑 → 𝜂)

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

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

This theorem is used by:  mpsyl4anc  856  swrdf1  14804  chnind  18801  idlmulssprm  21629  isprmidlc  21634  prmidlc  21635  qsidomlem2  21643  ucnima  24605  f1otrge  29455  mgcf1o  33611  gsumfs2d  33669  cycpmrn  33751  rlocisunit  33884  linds2eq  33983  rhmimaidl  34029  ply1unit  34154  lbsdiflsp0  34305  extdg1id  34345  3cubeslem1  43788  cantnftermord  44416  sineq0ALT  46013  cncfshift  46978  cncfperiod  46983  line2x  49960  inlinecirc02plem  49992