Theorem syl1111anc 841

Description: Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1377 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 511	. 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 838	1 ⊢ (𝜑 → 𝜂)

Syntax hints:   → wi 4   ∧ wa 395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  mpsyl4anc  843  chnind  18587  ucnima  24245  f1otrge  28940  swrdf1  33016  mgcf1o  33063  gsumfs2d  33122  cycpmrn  33204  linds2eq  33441  rhmimaidl  33492  idlmulssprm  33502  isprmidlc  33507  prmidlc  33508  qsidomlem2  33513  ply1unit  33635  lbsdiflsp0  33770  extdg1id  33810  3cubeslem1  43116  cantnftermord  43748  sineq0ALT  45363  cncfshift  46302  cncfperiod  46307