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  14792  chnind  18788  idlmulssprm  21616  isprmidlc  21621  prmidlc  21622  qsidomlem2  21630  ucnima  24592  f1otrge  29442  mgcf1o  33557  gsumfs2d  33615  cycpmrn  33697  rlocisunit  33830  linds2eq  33929  rhmimaidl  33975  ply1unit  34100  lbsdiflsp0  34251  extdg1id  34291  3cubeslem1  43674  cantnftermord  44306  sineq0ALT  45904  cncfshift  46853  cncfperiod  46858