Theorem syl2an2 558

Description: syl2an 283 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)

Hypotheses

Ref	Expression
syl2an2.1	⊢ (𝜑 → 𝜓)
syl2an2.2	⊢ ((𝜒 ∧ 𝜑) → 𝜃)
syl2an2.3	⊢ ((𝜓 ∧ 𝜃) → 𝜏)

Assertion

Ref	Expression
syl2an2	⊢ ((𝜒 ∧ 𝜑) → 𝜏)

Proof of Theorem syl2an2

Step	Hyp	Ref	Expression
1		syl2an2.1	. . 3 ⊢ (𝜑 → 𝜓)
2		syl2an2.2	. . 3 ⊢ ((𝜒 ∧ 𝜑) → 𝜃)
3		syl2an2.3	. . 3 ⊢ ((𝜓 ∧ 𝜃) → 𝜏)
4	1, 2, 3	syl2an 283	. 2 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜑)) → 𝜏)
5	4	anabss7 547	1 ⊢ ((𝜒 ∧ 𝜑) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  divalgmod  10327  gcdsupex  10349  gcdsupcl  10350  cncongr2  10486  isprm3  10500