Theorem syl122anc 1287

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 1283	1 ⊢ (𝜑 → 𝜁)

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:  divdiv32apd  9155  divcanap5d  9156  divcanap7d  9158  divdivap1d  9161  divdivap2d  9162  seq3coll  11317  cau3lem  11904  summodclem2a  12174  prodmodclem2a  12369  prmind2  12924  divnumden  13002  pceulem  13103  pcqmul  13112  pcqdiv  13116  pcexp  13118  pcaddlem  13148  pcbc  13160  abladdsub4  14209  ablpnpcan  14215  gsummptfidmadd  14252  lmodvs1  14763  blss2ps  15644  blss2  15645  blssps  15665  blss  15666  xmeter  15674  lgsdi  16373