Theorem syl122anc 1193

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)
syl122anc.6	⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) → 𝜁)

Assertion

Ref	Expression
syl122anc	⊢ (𝜑 → 𝜁)

Proof of Theorem syl122anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 ⊢ (𝜑 → 𝜓)
2		sylXanc.2	. 2 ⊢ (𝜑 → 𝜒)
3		sylXanc.3	. 2 ⊢ (𝜑 → 𝜃)
4		sylXanc.4	. . 3 ⊢ (𝜑 → 𝜏)
5		sylXanc.5	. . 3 ⊢ (𝜑 → 𝜂)
6	4, 5	jca 302	. 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂))
7		syl122anc.6	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) → 𝜁)
8	1, 2, 3, 6, 7	syl121anc 1189	1 ⊢ (𝜑 → 𝜁)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 930

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 932

This theorem is referenced by:  divdiv32apd  8437  divcanap5d  8438  divcanap7d  8440  divdivap1d  8443  divdivap2d  8444  seq3coll  10426  cau3lem  10726  summodclem2a  10989  prmind2  11594  divnumden  11666  blss2ps  12334  blss2  12335  blssps  12355  blss  12356  xmeter  12364