Theorem syl32anc 1224

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)
syl32anc.6	⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) → 𝜁)

Assertion

Ref	Expression
syl32anc	⊢ (𝜑 → 𝜁)

Proof of Theorem syl32anc

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 304	. 2 ⊢ (𝜑 → (𝜏 ∧ 𝜂))
7		syl32anc.6	. 2 ⊢ (((𝜓 ∧ 𝜒 ∧ 𝜃) ∧ (𝜏 ∧ 𝜂)) → 𝜁)
8	1, 2, 3, 6, 7	syl31anc 1219	1 ⊢ (𝜑 → 𝜁)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962

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

This theorem depends on definitions:  df-bi 116  df-3an 964

This theorem is referenced by:  ioom  10031  modifeq2int  10152  modaddmodup  10153  seq3f1olemqsum  10266  seq3f1o  10270  exple1  10342  leexp2rd  10447  facubnd  10484  permnn  10510  dfabsmax  10982  expcnvre  11265  dvdsadd2b  11529  dvdsmulgcd  11702  sqgcd  11706  bezoutr  11709  cncongr2  11774  pw2dvds  11833  hashgcdlem  11892  tgioo  12704