Theorem syldbl2 855

Description: Stacked hypotheseis implies goal. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypothesis

Ref	Expression
syldbl2.1	⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃))

Assertion

Ref	Expression
syldbl2	⊢ ((𝜑 ∧ 𝜓) → 𝜃)

Proof of Theorem syldbl2

Step	Hyp	Ref	Expression
1		syldbl2.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜓 → 𝜃))
2	1	com12 33	. 2 ⊢ (𝜓 → ((𝜑 ∧ 𝜓) → 𝜃))
3	2	anabsi7 684	1 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  elpredimg  6312  rexdif1en  9160  php  9206  fpwwe2lem4  10700  elfzoextl  13836  rprmdvdspow  34047  rprmdvdsprod  34048  constrmon  34358  nadddilem1  36939  aks4d1p3  43096  primrootsunit1  43115  primrootlekpowne0  43123  sticksstones1  43164  sticksstones11  43174  unitscyglem2  43214