Theorem a1dd 48

Description: Deduction introducing a nested embedded antecedent. (Contributed by NM, 17-Dec-2004.) (Proof shortened by O'Cat, 15-Jan-2008.)

Hypothesis

Ref	Expression
a1dd.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
a1dd	⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

Proof of Theorem a1dd

Step	Hyp	Ref	Expression
1		a1dd.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		ax-1 6	. 2 ⊢ (𝜒 → (𝜃 → 𝜒))
3	1, 2	syl6 33	1 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  exmidsssnc  4340  nnsub  9343  difelfzle  10541  facdiv  11176  facwordi  11178  faclbnd  11179  pfxccat3  11506  dvdsabseq  12614  divgcdcoprm0  12879  exprmfct  12916  prmfac1  12930  pockthg  13136  clwwlknonex2lem2  16679  bj-inf2vnlem2  16997