Theorem syl122anc 1280

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 306	. 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂))
7		syl122anc.6	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) → 𝜁)
8	1, 2, 3, 6, 7	syl121anc 1276	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:  divdiv32apd  8971  divcanap5d  8972  divcanap7d  8974  divdivap1d  8977  divdivap2d  8978  seq3coll  11072  cau3lem  11633  summodclem2a  11900  prodmodclem2a  12095  prmind2  12650  divnumden  12726  pceulem  12825  pcqmul  12834  pcqdiv  12838  pcexp  12840  pcaddlem  12870  pcbc  12882  abladdsub4  13859  ablpnpcan  13865  lmodvs1  14288  blss2ps  15088  blss2  15089  blssps  15109  blss  15110  xmeter  15118  lgsdi  15724