Theorem syl121anc 1254

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)
syl121anc.5	⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)

Assertion

Ref	Expression
syl121anc	⊢ (𝜑 → 𝜂)

Proof of Theorem syl121anc

Step	Hyp	Ref	Expression
1		sylXanc.1	. 2 ⊢ (𝜑 → 𝜓)
2		sylXanc.2	. . 3 ⊢ (𝜑 → 𝜒)
3		sylXanc.3	. . 3 ⊢ (𝜑 → 𝜃)
4	2, 3	jca 306	. 2 ⊢ (𝜑 → (𝜒 ∧ 𝜃))
5		sylXanc.4	. 2 ⊢ (𝜑 → 𝜏)
6		syl121anc.5	. 2 ⊢ ((𝜓 ∧ (𝜒 ∧ 𝜃) ∧ 𝜏) → 𝜂)
7	1, 4, 5, 6	syl3anc 1249	1 ⊢ (𝜑 → 𝜂)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980

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 982

This theorem is referenced by:  syl122anc  1258  tfisi  4624  tfrcllemsucfn  6420  sbthlemi6  7037  sbthlemi8  7039  div32apd  8858  div13apd  8859  expdivapd  10796  modfsummodlemstep  11639  pcqmul  12497  pcid  12518  pcneg  12519  pc2dvds  12524  pcz  12526  pcaddlem  12533  pcadd  12534  pcmpt2  12538  pcbc  12545  qexpz  12546  expnprm  12547  ennnfonelemg  12645  ssblex  14751