Theorem syl1111anc 840

Description: Four-hypothesis elimination deduction for an assertion with a singleton virtual hypothesis collection. Similar to syl112anc 1376 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 837	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  842  ucnima  24168  f1otrge  28799  swrdf1  32878  mgcf1o  32929  chnind  32937  gsumfs2d  32995  cycpmrn  33100  linds2eq  33352  rhmimaidl  33403  idlmulssprm  33413  isprmidlc  33418  prmidlc  33419  qsidomlem2  33424  ply1unit  33544  lbsdiflsp0  33622  extdg1id  33661  3cubeslem1  42672  cantnftermord  43309  sineq0ALT  44926  cncfshift  45872  cncfperiod  45877