Theorem syl33anc 1293

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)
sylXanc.6	⊢ (𝜑 → 𝜁)
syl33anc.7	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁)) → 𝜎)

Assertion

Ref	Expression
syl33anc	⊢ (𝜑 → 𝜎)

Proof of Theorem syl33anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. . 3 ⊢ (𝜑 → 𝜓)
2		sylXanc.2	. . 3 ⊢ (𝜑 → 𝜒)
3		sylXanc.3	. . 3 ⊢ (𝜑 → 𝜃)
4	1, 2, 3	3jca 1208	. 2 ⊢ (𝜑 → (𝜓 ∧ 𝜒 ∧ 𝜃))
5		sylXanc.4	. 2 ⊢ (𝜑 → 𝜏)
6		sylXanc.5	. 2 ⊢ (𝜑 → 𝜂)
7		sylXanc.6	. 2 ⊢ (𝜑 → 𝜁)
8		syl33anc.7	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂 ∧ 𝜁)) → 𝜎)
9	4, 5, 6, 7, 8	syl13anc 1280	1 ⊢ (𝜑 → 𝜎)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  strleund  13517  strext  13519  iscnp4  15456  cnpnei  15457  cnptopco  15460  cncnp  15468  cnptopresti  15476  lmtopcnp  15488  txcnp  15509  xmetrtri  15614  bl2in  15641  blhalf  15646  blssps  15665  blss  15666  upgriswlkdc  16818