Theorem syl221anc 1282

Description: Syllogism combined with contraction. (Contributed by NM, 11-Mar-2012.)

Hypotheses

Ref	Expression
sylXanc.1	⊢ (𝜑 → 𝜓)
sylXanc.2	⊢ (𝜑 → 𝜒)
sylXanc.3	⊢ (𝜑 → 𝜃)
sylXanc.4	⊢ (𝜑 → 𝜏)
sylXanc.5	⊢ (𝜑 → 𝜂)
syl221anc.6	⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏) ∧ 𝜂) → 𝜁)

Assertion

Ref	Expression
syl221anc	⊢ (𝜑 → 𝜁)

Proof of Theorem syl221anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 ⊢ (𝜑 → 𝜓)
2		sylXanc.2	. 2 ⊢ (𝜑 → 𝜒)
3		sylXanc.3	. . 3 ⊢ (𝜑 → 𝜃)
4		sylXanc.4	. . 3 ⊢ (𝜑 → 𝜏)
5	3, 4	jca 306	. 2 ⊢ (𝜑 → (𝜃 ∧ 𝜏))
6		sylXanc.5	. 2 ⊢ (𝜑 → 𝜂)
7		syl221anc.6	. 2 ⊢ (((𝜓 ∧ 𝜒) ∧ (𝜃 ∧ 𝜏) ∧ 𝜂) → 𝜁)
8	1, 2, 5, 6, 7	syl211anc 1277	1 ⊢ (𝜑 → 𝜁)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002

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 1004

This theorem is referenced by:  syl222anc  1287  vtocldf  2852  dmdcanapd  8983  exprecap  10819  fzowrddc  11200  xrbdtri  11808  2strbasg  13174  2stropg  13175  fnpr2o  13393  cnptoprest  14934  blssps  15122  blss  15123  metequiv2  15191  xmettx  15205  edgstruct  15885  usgr2v1e2w  16065